Where Did Prot Holo Come From?

JusticeRetroHunter.7684 · June 6, 2021

8 hours ago, Pimsley.3681 said:

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.

😄 I guess I should maybe try to simplify and summarize the conversation a little so that everyone knows what we we are talking about in plain English. He uses Set Theory to place the argument over the last few pages into a formal setting, and sets the record pretty straight on the argument. Set theory is real fundamental math stuff that I never really needed until now, so I'm glad he came into formalize it like this :

1) So the first thing is that if you have three objects or builds A, B and C, via injection shows that the expression A=B=C is just A, A ,A meaning that if the objects are equal they are the same object.

2) Second thing, is that you have our inequality from way earlier A>B>C>D. This inequality is formalized by Kolzar here, to describe the expression in terms of cardinality of a set of things. cardinality here is a bit hard to explain...but basically if you imagine each letter as a build, those builds have skills and traits and amulets, and all those subcomponents can be added up to give you some integer value. This value basically expresses how many things are in each build A, B C and D. The more things there are in each build, the more cardinality there is in total., and this determines the size of each build if that makes sense. So long as the cardinality of ABCD = ABCD at the end, A B C and D can take on any of their own individual values, and they can be arranged in terms of kernel size like A>B>C>D.

Now this last page, my argument is that ABC and D also describes the order of those things, which you can map one to one via the same injection as before, to the same set. Where the cardinality = ordinance. This is analogous to saying that A>B>C>D regardless of their size, will always be in an order. According to Kolzar right now, is that this can't be done in the model, but I don't truly understand why. Skipping over some details, This doesn't effect Kolzar's derivation in the end, it just leaves out an important detail about how fast the system is going from D to A.

3) Third thing, is that Kolzar very nicely describes an equilibrium point. The details on this is a bit gooey, but for now, the size of the fixed point is the same as the number of builds that act as attractors to becoming meta. If there are 100 attractors, and 1000 builds, then the meta will eventually be these 100 builds because those builds are in equilibrium. This is why Kolzar cares about the size of the fixed point. By size, he means how many builds act as attractors.

Last thing is this argument about computation time. It seems me and Kolzar and I are in two different camps on this, but basically my position is that there is a non-zero finite amount of time that it takes for someone to look at a skill, and make a decision on that skill...analyzing it's components, followed by giving that skill an ordinance among a system of other skills. You can imagine a trivial situation in which you have one game with 2 skills with only 1 parameter (damage maybe), and another a game with a million skills. How long do you think it will take a player to make a decision on which skill is the best? You can also imagine a another game, with 2 skills, but each skill has 1000 different parameters. Again you could ask how long would it take to arrive at an answer to that question. This argument above is basically a subject of Complex Computation, and how complex computation applies to gw2, and that the model setup by Kolzar needs to reflect that so we can get an idea for how long it takes for a system to strike equilibrium.

If the equilibrium is not large, then the computation time matters. If the equilibrium is large, then it's no biggie we got a large selection of builds that are viable right... but the question is what does it mean to have large equilibirums...it means skills have to be designed to conform to two extremes : Either Complete homogeneity, or a complete Rock Paper Scissors game. Neither solution seems logical to me for gw2. That's why I'm still trying to poke Kolzar's brain I hope he hasn't left yet.

Edited June 6, 2021 by JusticeRetroHunter.7684

Kolzar.9567 · June 6, 2021

11 hours ago, JusticeRetroHunter.7684 said:

Is there a reason why you are using a set valued function rather then a real valued function?

Ah this is actually quite an important question and the answer is shockingly simple, albeit not obvious (and unfortunately certainly not part of a regular undergraduate curriculum), every function is a correspondence (or a set valued function as it might be more familiar), but the reverse is not true. By imposing that your mapping is a function as opposed to a correspondence you are making an unjustified assumption. The difference can be highlighted via a simple Rock Paper Scissors(RPS) game where winner gets 1 dollar, loser pays one dollar and a draw is 0 dollar exchange. Now for the sake of the example I know the odds of you playing each of R,P,S through some means (for example statistical inference from your past plays against others or me). Now if I knew the odds of you throwing each of R, P or S is 1/3's (for the sake of the example) I can ask myself what can I do to maximize my expected dollar earnings. I can answer this question by simply comparing my returns from each of my pure strategies first. If I play rock, I will win 1/3 of the time, draw 1/3 of the time and lose 1/3rd of the time so my expected return from playing rock is ER(R)=1/3(+1)+1/3(0)+1/3(-1)=0. Now if you do the same calculation we see that the same is true for S and P as well. So my pure(in the sense that I am not randomizing like you) best response to your strategy of playing 1/3 each is a correspondance, PBR_Kolzar(1/3,1/3,1/3) ={R,P,S}. More notably if you were to allow me to mix as well, then I have infinitely many responses as I can essentially use a dice, a roulette wheel etc any kind of randomization. If you restricted me to a function on the other hand, I would always have to give an answer that is a singleton. (As a reminder a function maps something on the domain to a single object in the range, that is why for example the inverse of absolute value |x| is a correspondence) On the other hand though if you were to say you were playing R with certainty then my best response correspondence would yield a singleton so we lose nothing by dealing with a correspondence.

Now the comparison to physical phenomena vs people best responding might also be apt here. In physical phenomena, you have laws that govern such behavior and indifferences (like my equality of RPS) is not present even with noise you just have additional initial/boundary conditions. So to start the comparison lets define a best response operator as before, if it is BR_kolzar( x,y,z) where x,y,z are the probabilities you play R,P,S respectively (so they add up to 1 and are all weakly positive) and then operate infinitely often on this. Eventually we will realize BR_kolzar(1/3,1/3,1/3) contains the point (1/3,1/3,1/3) among a lot of (infinitely many) other things. But that means 1/3,1/3,1/3 is a fixed point of the single player best response operator. A curious observation here is that if we had considered, BR_kolzar(0.34,0.33,0.33) = {P}, that is it is a singleton. (the difference is minimal but it is strict, in fact you might say P>R=S in that case, where the > sign should be interpreted as " strictly preferred") You might notice that there is something funny going on here, as we only changed the odds of R by a really tiny bit but my best response reduced from infinitely many stuff to a single point. That is because unlike functions, continuity of correspondences are a little more delicate (you would need notions of upper/lower hemi-continuity) but it is very very natural. In physical phenomena, you assume there is some sort of law governing this continuity and it is sufficiently well behaved, in fact they are well behaved enough that you can talk about them in terms of derivatives. For example, lets think about the heat equation, we have a metal rod of some length, but we also heat it from some end and cool it from the other. So we can still ask will there be some point in the rod and some time after which that point doesn't change temperature. The heat at time t at length l would yield a partial differential equation, which means the solution is function (not a correspondance), albeit a multi dimensional one, and not only is it continuous (so no jumps), it doesn't even have kinks so that it is differentiable. As you previously suggested if we changed the shape of this metal rod to a sheet, a block or some fancy structure (that is by adding dimensions) calculation of the point with fixed temperature would become more complex, but the important assumptions are the differentiability and exogenous laws here. If we were to put those kinds of restrictions to my behavior in RPS, first I needed to only have singletons at every point say I am playing R against (1/3,1/3,1/3), I also wouldn't be allowed to jump to P, and I should be playing R for similar enough odds even though P or S might be superior. Therefore the assumption of functions limits the analysis significantly even for the simple RPS game. And actually since our best response correspondance lacks hemi-continuity in this game this is a very complex operation much more than say a 3d heat equation(from the perspective of complexity), despite kids playing it successfully. Furthermore the assumption of a function also limits the ability to consider the size of the fixed point, because it is always a singleton. Now to take it back to GW2, we can redo the analysis in a limited fashion albeit with some limitations. For example suppose we are going to duel and I know you are going to play say some build of necromancer. (or some distribution again I acquired from observation), I ask myself what are viable counters to the distribution of builds you are going to play, if there is more than 1 answer for me (this now has to include my ability to play said classes too, say I can counter your necro with a DD or ele but I suck at playing any kind of Ranger, although it might be a viable counter), we have to use a correspondance. Then you can ask too what are the distribution of builds Kolzar can play given that he is going to best respond to his expectation of my builds. And curiously enough we can define the two dimensional best response correspondence (where we simultaneously best respond to each other) and show that if everything is finite enough there is going to be a fixed point, or an equilibrium. (In fact the existence of this is the celebrated Nash equilibrium concept, although a pure mathematician would call it a simple application of Kakutani's fixed point theorem, which really is intermediate value theorem for correspondences) It might be computationally very complex the best responses are correspondences but unlike computers humans deal with such issues fairly easily at least in terms of an approximate sense, much faster than computers actually. In fact this is a thriving area of research in the intersection of neuroscience and experimental economics.

Edited June 6, 2021 by Kolzar.9567
clarification

Kolzar.9567 · June 6, 2021

3 hours ago, JusticeRetroHunter.7684 said:

cardinality here is a bit hard to explain

I apologize for the confusion here, cardinality in a finite set is just the number of elements in that set. So the set {a,b,c,d} has cardinality 4 and so on. Adding to what I earlier posted E({a,b,c,d}) = {q,w,e,r,t,y} should be interpreted as if the FOTM builds are {a,b,c,d} their counters, are the builds {q,w,e,r,t,y}. You are in a stable meta (equilibrium/fixed point) when you have some collection say {a,b,x,y} such that E({a,b,x,y})={a,b,x,y}. This meta is diverse if the cardinality of the set {a,b,x,y} is large. And formally, how many different FOTM iterations you have is the complexity of the fixed point, that is E(E(E(E (x,y,z))))))=S and E(S)=S. I keep using cardinality since it is the formal term for any set (so you can think about the cardinality of real numbers or natural numbers for example and actually compare them, whereas asking the number of elements only gives you infinity).

And just to add on I think of E as the appropriate endogenous correspondence, which kind of depends on how players counter other builds as I tried to illustrate in the earlier post of how to counter rock in RPS.

Edited June 6, 2021 by Kolzar.9567

JusticeRetroHunter.7684 · June 7, 2021

11 hours ago, Kolzar.9567 said:

Ah this is actually quite an important question and the answer is shockingly simple, albeit not obvious (and unfortunately certainly not part of a regular undergraduate curriculum), every function is a correspondence (or a set valued function as it might be more familiar), but the reverse is not true. By imposing that your mapping is a function as opposed to a correspondence you are making an unjustified assumption. The difference can be highlighted via a simple Rock Paper Scissors(RPS) game where winner gets 1 dollar, loser pays one dollar and a draw is 0 dollar exchange. Now for the sake of the example I know the odds of you playing each of R,P,S through some means (for example statistical inference from your past plays against others or me). Now if I knew the odds of you throwing each of R, P or S is 1/3's (for the sake of the example) I can ask myself what can I do to maximize my expected dollar earnings. I can answer this question by simply comparing my returns from each of my pure strategies first. If I play rock, I will win 1/3 of the time, draw 1/3 of the time and lose 1/3rd of the time so my expected return from playing rock is ER(R)=1/3(+1)+1/3(0)+1/3(-1)=0. Now if you do the same calculation we see that the same is true for S and P as well. So my pure(in the sense that I am not randomizing like you) best response to your strategy of playing 1/3 each is a correspondance, PBR_Kolzar(1/3,1/3,1/3) ={R,P,S}. More notably if you were to allow me to mix as well, then I have infinitely many responses as I can essentially use a dice, a roulette wheel etc any kind of randomization. If you restricted me to a function on the other hand, I would always have to give an answer that is a singleton. (As a reminder a function maps something on the domain to a single object in the range, that is why for example the inverse of absolute value |x| is a correspondence) On the other hand though if you were to say you were playing R with certainty then my best response correspondence would yield a singleton so we lose nothing by dealing with a correspondence.

Now the comparison to physical phenomena vs people best responding might also be apt here. In physical phenomena, you have laws that govern such behavior and indifferences (like my equality of RPS) is not present even with noise you just have additional initial/boundary conditions. So to start the comparison lets define a best response operator as before, if it is BR_kolzar( x,y,z) where x,y,z are the probabilities you play R,P,S respectively (so they add up to 1 and are all weakly positive) and then operate infinitely often on this. Eventually we will realize BR_kolzar(1/3,1/3,1/3) contains the point (1/3,1/3,1/3) among a lot of (infinitely many) other things. But that means 1/3,1/3,1/3 is a fixed point of the single player best response operator. A curious observation here is that if we had considered, BR_kolzar(0.34,0.33,0.33) = {P}, that is it is a singleton. (the difference is minimal but it is strict, in fact you might say P>R=S in that case, where the > sign should be interpreted as " strictly preferred") You might notice that there is something funny going on here, as we only changed the odds of R by a really tiny bit but my best response reduced from infinitely many stuff to a single point. That is because unlike functions, continuity of correspondences are a little more delicate (you would need notions of upper/lower hemi-continuity) but it is very very natural. In physical phenomena, you assume there is some sort of law governing this continuity and it is sufficiently well behaved, in fact they are well behaved enough that you can talk about them in terms of derivatives. For example, lets think about the heat equation, we have a metal rod of some length, but we also heat it from some end and cool it from the other. So we can still ask will there be some point in the rod and some time after which that point doesn't change temperature. The heat at time t at length l would yield a partial differential equation, which means the solution is function (not a correspondance), albeit a multi dimensional one, and not only is it continuous (so no jumps), it doesn't even have kinks so that it is differentiable. As you previously suggested if we changed the shape of this metal rod to a sheet, a block or some fancy structure (that is by adding dimensions) calculation of the point with fixed temperature would become more complex, but the important assumptions are the differentiability and exogenous laws here. If we were to put those kinds of restrictions to my behavior in RPS, first I needed to only have singletons at every point say I am playing R against (1/3,1/3,1/3), I also wouldn't be allowed to jump to P, and I should be playing R for similar enough odds even though P or S might be superior. Therefore the assumption of functions limits the analysis significantly even for the simple RPS game. And actually since our best response correspondance lacks hemi-continuity in this game this is a very complex operation much more than say a 3d heat equation(from the perspective of complexity), despite kids playing it successfully. Furthermore the assumption of a function also limits the ability to consider the size of the fixed point, because it is always a singleton. Now to take it back to GW2, we can redo the analysis in a limited fashion albeit with some limitations. For example suppose we are going to duel and I know you are going to play say some build of necromancer. (or some distribution again I acquired from observation), I ask myself what are viable counters to the distribution of builds you are going to play, if there is more than 1 answer for me (this now has to include my ability to play said classes too, say I can counter your necro with a DD or ele but I suck at playing any kind of Ranger, although it might be a viable counter), we have to use a correspondance. Then you can ask too what are the distribution of builds Kolzar can play given that he is going to best respond to his expectation of my builds. And curiously enough we can define the two dimensional best response correspondence (where we simultaneously best respond to each other) and show that if everything is finite enough there is going to be a fixed point, or an equilibrium. (In fact the existence of this is the celebrated Nash equilibrium concept, although a pure mathematician would call it a simple application of Kakutani's fixed point theorem, which really is intermediate value theorem for correspondences) It might be computationally very complex the best responses are correspondences but unlike computers humans deal with such issues fairly easily at least in terms of an approximate sense, much faster than computers actually. In fact this is a thriving area of research in the intersection of neuroscience and experimental economics.

So this is a great comment, and I took some time to analyze it in detail. I think we both now said this, but human computation acts like an algorithm, that approximates the answer in the complexity space of the system. Many players doing these approximate computations in parallel, brings the problem out of NP and into P, where someone can propose that Build A is the best build to play as a response to any other build, but to verify that A is the best build, requires enormous computation, and the algorithm, being human decision making being run in parallel, can approximately verify that answer. I see this as a flaw in the design of the game, in that the game should be in the space of NP and not solvable in P. Chess easily trumps gw2 in terms of complex computation and I think that says a lot about gw2 and it's game design.

In all of my arguments here, I encourage treating the fixed point of the game as being just 1 build, where build A whatever it is, is the best build. This way, we just assume that the equilibrium is small, because to verify that the game may have a large equilibrium is a complex computation, and it requires the game to be played by 1000 or so players in parallel in order to verify that answer. Looking at the state of the game right now, where there are 6 meta builds, implies that the equilibrium is indeed very small.

Now the above is from the perspective of looking at diversity as an equilibrium point. However, by looking at diversity from the perspective of complex computation time, then whatever that equilibrium point is...well it doesn't really matter...since players will be forever trying out different combinations if the computational complexity of the game was in NP...wouldn't you agree?

Edited June 7, 2021 by JusticeRetroHunter.7684

Crozame.4098 · June 7, 2021

lol, people still argue with our chaos theory god. This guy is funny. At the very basic level, he thinks nerfing the numbers will make the game less diverse... emmm, we have 9 classes, each class has different skills, how can change numbers makes the game less diverse, while animation, role of the build, CD etc still differs a lot.

This guy is just a nerd who think he knows the stuff, but clearly cannot explain properly, and each argument has its logical errors. Its just funny~

Kolzar.9567 · June 7, 2021

9 hours ago, JusticeRetroHunter.7684 said:

So this is a great comment, and I took some time to analyze it in detail. I think we both now said this, but human computation acts like an algorithm, that approximates the answer in the complexity space of the system.

I think this is the heart of the difference in our views, human computation at least in my experience/knowledge, does not actually fully obey the norms of complexity as we classify them. That was why I was trying to give the examples of RPS, Cournot and Walrasian Equilibria. By their very nature these environments are very complex, just out of the fact that that humans optimizing must use correspondences (remember this is without loss as every function is a correspondence) and if these correspondances are not well behaved, like in the case of RPS, or Walrasian Equilibrium (this is the usual supply equals demand, but with multiple goods so you don't draw a single supple and demand curve, but have many of them clearing simultaneously, basically the simplest barter equilibrium you can think of) are very complex problems, yet humans do remarkably well: kids play RPS, we don't see bouts of hyperinflation or market failure (although curiously enough the subprime mortgage crisis was a failure of the market makers not understanding basic probability and incorporating it into their pricing of mortgage risk) The reason a kid might throw R,P,S with equal odds is not because they spent eons doing the calculation (which is approximately how long it will take) to find the equilibrium strategies, but something a little extra unique to humans.

However, this is not to say complexity cannot have a part, the issue of complexity arises when your own strategies are hard to evaluate for yourself. For example in RPS, you have three strategies R,P,S, and you know exactly what they do. Randomizing is also easy, as you simply take expectations over a random variable with a very simple support (the outcome space of RPS, RR, RP, etc). In chess on the other hand, a strategy is a "complete contingent plan of action" (this is indeed the formal definition), which means for every contingency you need to have a dictated action. Now by nature, since the game is sequential it ought to be easier, imagine playing RPS where you get to play after seeing what I picked. But the horizon is too large so that you actually cannot compute the expected return from playing a strategy over another. Normally since the game is sequential you should be able to solve it via backward induction, but the sheer complexity of the game tree prevents you from doing so. That is why both people and computers approximate starting from a given board position. (There was a curious experiment where they threw random board positions to GM's in chess and asked them to play the game to there, if it was a completely random position GM's were quite stumped, yet if it was from a position in an actual game they could reconstruct it). In guild wars on the other hand I don't think this is a problem, if I know my build, and I have a knowledge of the other classes, I can evaluate the odds of me coming out on top in any encounter (including my own skill level and my expectation of the population). That is why when to save your stunsbreaks/condi cleanses etc when facing a certain class. Now if the game was too complex in the sense that I could not evaluate when to stunbreak then I would agree complexity is the issue, but I just don't think it is the case for GW2. This also the part where it becomes a more subjective opinion though, as I don't think anyone did some formal analysis of GW2 (one might consider this thread the first attempt 🙂 ). For the other examples such as Walrasian equilibrium, Cournot, RPS and even chess (do look up zermelo's theorem although the wikipedia article is a little dissatisfactory) there is a lot of research already done, both on their complexity and experiments to actually measure human reaction.

Crozame.4098 · June 7, 2021

7 minutes ago, Kolzar.9567 said:

I think this is the heart of the difference in our views, human computation at least in my experience/knowledge, does not actually fully obey the norms of complexity as we classify them. That was why I was trying to give the examples of RPS, Cournot and Walrasian Equilibria. By their very nature these environments are very complex, just out of the fact that that humans optimizing must use correspondences (remember this is without loss as every function is a correspondence) and if these correspondances are not well behaved, like in the case of RPS, or Walrasian Equilibrium (this is the usual supply equals demand, but with multiple goods so you don't draw a single supple and demand curve, but have many of them clearing simultaneously, basically the simplest barter equilibrium you can think of) are very complex problems, yet humans do remarkably well: kids play RPS, we don't see bouts of hyperinflation or market failure (although curiously enough the subprime mortgage crisis was a failure of the market makers not understanding basic probability and incorporating it into their pricing of mortgage risk) The reason a kid might throw R,P,S with equal odds is not because they spent eons doing the calculation (which is approximately how long it will take) to find the equilibrium strategies, but something a little extra unique to humans.

However, this is not to say complexity cannot have a part, the issue of complexity arises when your own strategies are hard to evaluate for yourself. For example in RPS, you have three strategies R,P,S, and you know exactly what they do. Randomizing is also easy, as you simply take expectations over a random variable with a very simple support (the outcome space of RPS, RR, RP, etc). In chess on the other hand, a strategy is a "complete contingent plan of action" (this is indeed the formal definition), which means for every contingency you need to have a dictated action. Now by nature, since the game is sequential it ought to be easier, imagine playing RPS where you get to play after seeing what I picked. But the horizon is too large so that you actually cannot compute the expected return from playing a strategy over another. Normally since the game is sequential you should be able to solve it via backward induction, but the sheer complexity of the game tree prevents you from doing so. That is why both people and computers approximate starting from a given board position. (There was a curious experiment where they threw random board positions to GM's in chess and asked them to play the game to there, if it was a completely random position GM's were quite stumped, yet if it was from a position in an actual game they could reconstruct it). In guild wars on the other hand I don't think this is a problem, if I know my build, and I have a knowledge of the other classes, I can evaluate the odds of me coming out on top in any encounter (including my own skill level and my expectation of the population). That is why when to save your stunsbreaks/condi cleanses etc when facing a certain class. Now if the game was too complex in the sense that I could not evaluate when to stunbreak then I would agree complexity is the issue, but I just don't think it is the case for GW2. This also the part where it becomes a more subjective opinion though, as I don't think anyone did some formal analysis of GW2 (one might consider this thread the first attempt 🙂 ). For the other examples such as Walrasian equilibrium, Cournot, RPS and even chess (do look up zermelo's theorem although the wikipedia article is a little dissatisfactory) there is a lot of research already done, both on their complexity and experiments to actually measure human reaction.

I mean, why you are also making things unnecessarily complicated? What is the key assumption in economics -- especially in game theory, btw, is quite common in undergraduate curriculums. Does that assumption hold in GW2 game play?

Also, since you mentioned experimental economics, then just a few example: do people give positive amounts in a Dictator game? Do subjects contribution positive amount in publics good game? What is the average guess in the Beauty contest game? Do subjects bid the equilibrium in second prize auctions? Do subjects suffer the winners curse in first prize auctions?

Kolzar.9567 · June 7, 2021

49 minutes ago, Crozame.4098 said:

I mean, why you are also making things unnecessarily complicated? What is the key assumption in economics -- especially in game theory, btw, is quite common in undergraduate curriculums. Does that assumption hold in GW2 game play?

I have made 0 assumptions on how people evaluate their outcomes more notably their preferences that is why left that part undefined as I repeatedly mentioned. If you are somehow going to go with the argument that you need infinite levels of rationality, that simply is not just necessary to describe the rest of the environment. You can indeed have notions of equilibrium and various forms of bounded rationality as you wish which will indeed change your prediction of the outcome of the game, but does not need to interact with the environment. Notably you can have people with varying levels/kinds of bounded rationality interact in the same environment. Or in fact you can try to both theoretically and experimentally try to measure how much of an impact such bounded rationality will have but again that does not invalidate a formal description of the environment.

Suppose you are trying to plan your retirement and for some reason you are present biased, the lack of rationality (or more precisely time-inconsistency) does not prevent anyone from a describing how your savings would evolve, what kind of paths would be appropriate etc, now if you were buying your retirement assets from again some lets say future biased financial intermediary, again describing the environment can be done simply. In fact if there are multiple such people we can even define measures of say inequality in savings etc again, which is separate from how irrational said people are. Predicting the path of play would require assumptions on the behavior of these agents (which you can try to do experiments on or somehow infer it from existing data), but that is not something I have done. I have simply stated what I think was the right notion of diversity, and proposed two models of describing the environment, one with a distribution, one without. The rest of the argument was to highlight why I the notion of diversity I came up with is appropriate basing off of very very simple games, and how it is unrelated to the complexity of the problem at hand.

I certainly appreciate criticisms to the modelling choice but I don't understand what you are refuting with the references to experimental behavior not matching mathematical analysis in the various games you are mentioning. I am not making any predictions on play, I am simply trying to propose a model where there is well defined notion of diversity, I don't see how people's irrationality or preferences in general invalidates this notion. You can say that the notion and the entire modelling choice is wrong, but I am failing to see how it is related to the examples you gave in experimental economics. People with bounded rationality still play dictator games, beauty contests etc, they are not stupified (as if unable to compute what to do because of complexity) and if they repeat the game often enough they do converge to some focal behavior. Now this focal behavior might indeed be related to their form of bounded rationality, but if we needed understand exactly what kind of bounded rationality is there we need a model that allows us to pinpoint that exactly. It does not invalidate how different rounds will go through, or the game or its rules.

If you are somehow irritated by the length of the posts this is because I am trying to be as simple as possible giving examples I hope people have heard and so that people can contribute their own opinions and tell me where I am wrong in my notion of diversity, clearly I only have a certain level of knowledge and a certain perspective, and I need to be both somehow precise yet understandable when I am trying to seek out opinions of a general public. I disagree with the relationship between complexity and diversity that the other poster posed, so I try to illustrate the differences further with various very simple examples. I try to explain in detail why I think this relationship is not the correct one, and to do that I try to give precise, yet hopefully understandable model of my logic. I believe this is more useful than having cryptic questions posed, but that is my very subjective opinion.

Edited June 7, 2021 by Kolzar.9567

JusticeRetroHunter.7684 · June 7, 2021

1 hour ago, Crozame.4098 said:

lol, people still argue with our chaos theory god. This guy is funny. At the very basic level, he thinks nerfing the numbers will make the game less diverse... emmm, we have 9 classes, each class has different skills, how can change numbers makes the game less diverse, while animation, role of the build, CD etc still differs a lot.

This guy is just a nerd who think he knows the stuff, but clearly cannot explain properly, and each argument has its logical errors. Its just funny~

Me and Kolzar aren't really arguing. He's explaining a model of gw2 from a very formalized perspective. In fact I agree with this model with just one exception and we've discussed this exception in detail now.

Quote

I mean, why you are also making things unnecessarily complicated? What is the key assumption in economics -- especially in game theory, btw, is quite common in undergraduate curriculums. Does that assumption hold in GW2 game play?

What Kolzar is doing isn't overly complicating it either. He's describing the system in terms of sets, he's mapping these sets to a phase space, which shows you the systems evolution in time, and you derive solutions from the phase space so that you can predict it's behavior. This is fundamental analysis to how anyone would describe any system in every mathematical or scientific field. Chaos theory is no different...Chaos theory analyzes the phase space of non-linear systems...gw2 is a non-linear system, economics is a non-linear system...really all systems in the world are non linear unless you're doing something as basic as a 2 body problem...which is still chaotic just LESS chaotic and can be meaningfully approximated as linear.

Edited June 7, 2021 by JusticeRetroHunter.7684

Kuma.1503 · June 7, 2021

I'm going to level with you guys here. I just woke up after a late night of working. My brain is not operating at 100% capacity. Even if it was, and I was able to overclock it to work beyond it's maximum capacity, I still would not be able to comprehend a word of what's been said here.

I tried. I read the through each paragraph. My eyes took in the visual information of words on a screen. The words passed into the deep dark cavern that is my skull, sloshed around for a bit, then leaked out my ears like run-off.

This convo is like two Asura arguing with each other, using big complex words, then acting flabbergasted when the big dumb Norn doesn't comprehend.

"I cannot put it more simply" They insist, as the Norn stares on with a blank expression. He probably eats rocks and draws pictures in caves in his spare time. It's a miracle his brain doesn't explode when he hears the word "intermediary".

The Norn is me. For the sake of my mental health, I'm going to go back to eating rocks.

Edited June 7, 2021 by Kuma.1503

Crozame.4098 · June 7, 2021

1 hour ago, JusticeRetroHunter.7684 said:

What Kolzar is doing isn't overly complicating it either. He's describing the system in terms of sets, he's mapping these sets to a phase space, which shows you the systems evolution in time, and you derive solutions from the phase space so that you can predict it's behavior. This is fundamental analysis to how anyone would describe any system in every mathematical or scientific field. Chaos theory is no different...Chaos theory analyzes the phase space of non-linear systems...gw2 is a non-linear system, economics is a non-linear system...really all systems in the world are non linear unless you're doing something as basic as a 2 body problem...which is still chaotic just LESS chaotic and can be meaningfully approximated as linear.

You dont even understand what I was referring...

Oh please, he is applying game theory. Again, the issue is how he applied it to the game. To reach Nash equilibrium, we need rationality and for some solution concepts, need common knowledge of rationality. But none of the players are rational, we can say some of them are more sophisticated than others. But no one is rational.

And the PSR, which is a simultaneous game where each player's strategic set has a cardinality of 3. Not applicable... In games, skills levels are different, builds are different, and the duelling is both simultaneous and sequential, and given the different incentives people have in those games, you cannot even define the game first, let alone trying to solve the NE...

Kolzar.9567 · June 7, 2021

14 minutes ago, Crozame.4098 said:

And the PSR, which is a simultaneous game where each player's strategic set has a cardinality of 3. Not applicable... In games, skills levels are different, builds are different, and the duelling is both simultaneous and sequential, and given the different incentives people have in those games, you cannot even define the game first, let alone trying to solve the NE...

I am slightly puzzled by this comment, as I mentioned before there are notions of equilibria for various forms of bounded rationality, maybe you want to use Berk-Nash if you think people are learning in a misspecified form maybe you want to impose k level rationality, or if you think they are time inconsistent maybe you want to use sophistication as you eluded to. I gave the examples of RPS because other users mentioned it, I gave the examples of Walrasian equilibrium and Cournot because those are examples I thought people have heard of. If you really want to formalize PvP balance properly indeed at the very least you should think of a stochastic game with randomized matching that has both simultaneous and sequential parts and even some incomplete information, beside the preferences of the players, and think of the state as some distribution of the meta builds. Notice I still leave how you would actually play in an encounter undefined, I am just thinking of build diversity here, that is the process governing the distribution of meta builds. Furthermore you need to incorporate player knowledge, skill and even preferences; maybe I just think playing weaver is the coolest thing ever and I don't want to necessarily win and I can't use any utilities because they are too far in my keyboard and I can't understand other classes and I am not very forward looking. Still what remains is that given my characteristics I will try to optimize to some extent, which might be distinctly different from how you would do so, and maybe use a dagger instead of sword. In the end there will be an outcome that I am part of, which you can formally describe as part of a stochastic process, as I will be some tiny dot in the support of your process. Since everybody is jointly maximizing something the resulting behavior is still going to have some structure. Making predictions on path of play does indeed require assumptions on behavior, knowledge and rationality which I have not attempted to do. Again to give a very very simple example, suppose we play RPS 3 times, I like to play rock no matter what, you may or may not know this. The path of play could be (S,R), (P,R) (P,R). Do you need to know how happy I am to play rock? No, you can still at least tell it is going to be some sequence resulting from my boundedly rational maximization. And once again I have never made any predictions on the path of play. That is I never said we are using Nash, Berk-Nash, Sequential Eq etc, I just said it comes from an optimization on the player base and it will be a correspondance. It will reach a stable meta, when the player base does not want to switch given their assessment of the environment. Just to really make the point clear, I have not made any assumptions on behavior, and I certainly do not predict the path of play, or impose any solution concept let alone calculating something like Nash equilibria without any justification as you mentioned.

On a beside point, clearly any model you can come up with is flawed as it is an attempt to map reality into a tractable form, it is necessarily going to omit some aspects. Saying game theory is useless based on this, at least in my opinion, is not correct. The goal of a modelling exercise is to have a structure to think of such issues and discuss to improve our understanding of the dynamics. Game theory itself is not a perfect replica of life either, as both you and I mentioned earlier infinite levels of rationality is not realistic, if you want to somehow add probabilities, IIA is even more problematic and as you go along you get more and more problematic structures. But the goal is to first admit the shortcomings of a model and try to see how much it can explain. That is why you describe it clearly so people can refute the assumptions you make through the process. However if you think it is good enough in a sense it serves as a useful tool, for example you do not completely dispose of Newtonian physics because there is quantum mechanics, an approximation it is good enough to have some understanding.

Sigmoid.7082 · June 7, 2021

I wondered why this thread got so long....makes sense now.

Crozame.4098 · June 7, 2021

1 hour ago, Kolzar.9567 said:

However if you think it is good enough in a sense it serves as a useful tool, for example you do not completely dispose of Newtonian physics because there is quantum mechanics, an approximation it is good enough to have some understanding.

Thats the point, neither chaos theory (based on his explanation) nor game theory is the tool for balancing the game. ANET has data, that alone should be enough.

ok, let me elaborate a bit more. I remember a while ago I had a quite long discussion with the Chaos theory guy. At a certain point, I asked: ok, assume that you are right and simply adjusting numbers does not work, then what's your solution? He/she replied, that's a complicated question and I have no good answer.

I mean, there is no way you can arugue with someone like that~ Its like arguing whether we live in the Matrix or not~

Edited June 7, 2021 by Crozame.4098

JusticeRetroHunter.7684 · June 7, 2021

4 minutes ago, Crozame.4098 said:

Thats the point, neither chaos theory (based on his explanation) nor game theory is the tool for balancing the game. ANET has data, that alone should be enough.

Tell me, what do you think chaos theory is.

Crozame.4098 · June 7, 2021

12 minutes ago, JusticeRetroHunter.7684 said:

Tell me, what do you think chaos theory is.

tell me how to realistically make the game more enjoyable by not balancing the numbers? IN other words, apply concretely how the chaos theory can balance. Note: concrete.

And I emphasise again, I am not questioning the Chao theory, rather how you apply it for balance.

Edited June 7, 2021 by Crozame.4098

JusticeRetroHunter.7684 · June 7, 2021

30 minutes ago, Crozame.4098 said:

tell me how to realistically make the game more enjoyable by not balancing the numbers? IN other words, apply concretely how the chaos theory can balance. Note: concrete.

And I emphasise again, I am not questioning the Chao theory, rather how you apply it for balance.

You can't answer my question, with a question... Answer the question i asked you first.

Crozame.4098 · June 7, 2021

1 hour ago, JusticeRetroHunter.7684 said:

You can't answer my question, with a question... Answer the question i asked you first.

ok, I think Chaos theory is science, but the way you apply it to GW2's balance does not make sense.

So now: tell me how to realistically make the game more enjoyable by not balancing the numbers? IN other words, apply concretely how the chaos theory can balance. Note: concrete.

Kolzar.9567 · June 7, 2021

2 hours ago, Crozame.4098 said:

Thats the point, neither chaos theory (based on his explanation) nor game theory is the tool for balancing the game. ANET has data, that alone should be enough.

I am sorry but I think you are missing something here, any kind of data does need to have model to make sense out of it. That is literally how you do any sort of econometric analysis, otherwise all you will have is random correlations without any causality. You seem to be confusing a modelling exercise and imposing a solution concept which I have not done. Since there is no solution that I imposed I am not even understanding why you are claiming that I am applying some solution from game theory, I am simply describing a strategic environment, simplified enough to retain what I think is relevant so that there is possibility of meaningful discussion without alienating the general public. This last point is important as I am trying to be as simple as possible with as many examples as possible and relating to topics you might have heard of to allow for contribution from everyone.
To give yet another simple example you can consider traffic, people are going from their origins to their destinations which is completely their own preference, the are some laws that they may not be fully aware of or obey and they have their own driving capabilities that makes them choose different routes. Nonetheless if you are trying to get a sense of the traffic at an intersection, or the number of cars at road at a given time, you can make the very basic assumption that people don't want to get into a crash and try to get to their destination in a way that is maximizing their their preferences (not utility as they might be too irrational to have utility representation but a binary relationship that might even be incomplete or intransitive). Now at this point you can make further assumptions on peoples behavior and try to predict behavior using a solution concept. There assumptions need to be refutable, and if they are not solidly grounded, it will invalidate your predictions, but not necessarily your description of the environment. Alternatively you can assume some form of noise, identify what is endogenous and what is not, and try to make statistical inference using the base model. Or you can do something in between maybe assume some parametric behavior but correct it using data. What I am not seeing from your argument is what is preventing you from saying on the highway from x to y, at date t, there were n cars and these people were trying to go to their destination to the best of their ability. In fact the comparative statics exercises you can do with respect to the model is going to tell you how to make adjustments. Then actually you can ask and say if I were to widen this road a bit, there would be fewer cars so less congestion etc, regardless of rationality without imposing further solution concept. Just to see the fallacy of using data without a model with a rather ridiculous example, I see that as I go further north from my home, I see there are more cars, so traffic is increasing as I go north. Is there a valid correlation you can get from data, certainly, is the causality there? Not really. Certainly NJ should not increase their sizes of their roads based on how north it is from my home. So to hone in on the point, you must have model to balance stuff, or to use any kind of data. Solution concepts you impose on a model is distinctly different problem.

The only basic premise I had was there is a set of builds that are FOTM, people will do their own best to find builds that counter them which will give you an E(Fotm). If you are in a set such that Fotm=E(Fotm) you are in a stable meta. You don't need much here other than people are maximizing their preferences and given their own beliefs (again not utilities because irrationality, bounded rationality are all ok here, they might even be learning their preferences as they go, and even have misspecifications). Your solution concept and assumptions on people will provide you with a characterization of E, which I never did and never even attempted to. As I previously mentioned you can incorporate proportions to this, so you have process, a set and a distribution, and once again once people do their best given their capabilities maybe it will stabilize. You can disagree on this premise itself, but given the very very few assumptions I made I am not understanding your source of disagreement in the modelling exercise.

Edit: On arguing with people, at least in my opinion you cannot have an argument unless you actually try to understand people and be respectful. Again in my opinion people's understanding will certainly vary based on their background and knowledge, that is why despite my disagreement with the other poster and you, I try to explain my position repeatedly using various examples so we can at least understand each other to agree to disagree. Having more or less knowledge on a topic does not invalidate a persons ability to think and if I am using knowledge other people don't it is my responsibility to convey that knowledge to hear their opinion. Not necessarily dismiss people because I know some obscure bit of knowledge that they don't.

Edited June 7, 2021 by Kolzar.9567

Crozame.4098 · June 7, 2021

1 hour ago, Kolzar.9567 said:

I am sorry but I think you are missing something here, any kind of data does need to have model to make sense out of it. That is literally how you do any sort of econometric analysis, otherwise all you will have is random correlations without any causality. You seem to be confusing a modelling exercise and imposing a solution concept which I have not done. Since there is no solution that I imposed I am not even understanding why you are claiming that I am applying some solution from game theory, I am simply describing a strategic environment, simplified enough to retain what I think is relevant so that there is possibility of meaningful discussion without alienating the general public. This last point is important as I am trying to be as simple as possible with as many examples as possible and relating to topics you might have heard of to allow for contribution from everyone.
To give yet another simple example you can consider traffic, people are going from their origins to their destinations which is completely their own preference, the are some laws that they may not be fully aware of or obey and they have their own driving capabilities that makes them choose different routes. Nonetheless if you are trying to get a sense of the traffic at an intersection, or the number of cars at road at a given time, you can make the very basic assumption that people don't want to get into a crash and try to get to their destination in a way that is maximizing their their preferences (not utility as they might be too irrational to have utility representation but a binary relationship that might even be incomplete or intransitive). Now at this point you can make further assumptions on peoples behavior and try to predict behavior using a solution concept. There assumptions need to be refutable, and if they are not solidly grounded, it will invalidate your predictions, but not necessarily your description of the environment. Alternatively you can assume some form of noise, identify what is endogenous and what is not, and try to make statistical inference using the base model. Or you can do something in between maybe assume some parametric behavior but correct it using data. What I am not seeing from your argument is what is preventing you from saying on the highway from x to y, at date t, there were n cars and these people were trying to go to their destination to the best of their ability. In fact the comparative statics exercises you can do with respect to the model is going to tell you how to make adjustments. Then actually you can ask and say if I were to widen this road a bit, there would be fewer cars so less congestion etc, regardless of rationality without imposing further solution concept. Just to see the fallacy of using data without a model with a rather ridiculous example, I see that as I go further north from my home, I see there are more cars, so traffic is increasing as I go north. Is there a valid correlation you can get from data, certainly, is the causality there? Not really. Certainly NJ should not increase their sizes of their roads based on how north it is from my home. So to hone in on the point, you must have model to balance stuff, or to use any kind of data. Solution concepts you impose on a model is distinctly different problem.

The only basic premise I had was there is a set of builds that are FOTM, people will do their own best to find builds that counter them which will give you an E(Fotm). If you are in a set such that Fotm=E(Fotm) you are in a stable meta. You don't need much here other than people are maximizing their preferences and given their own beliefs (again not utilities because irrationality, bounded rationality are all ok here, they might even be learning their preferences as they go, and even have misspecifications). Your solution concept and assumptions on people will provide you with a characterization of E, which I never did and never even attempted to. As I previously mentioned you can incorporate proportions to this, so you have process, a set and a distribution, and once again once people do their best given their capabilities maybe it will stabilize. You can disagree on this premise itself, but given the very very few assumptions I made I am not understanding your source of disagreement in the modelling exercise.

Edit: On arguing with people, at least in my opinion you cannot have an argument unless you actually try to understand people and be respectful. Again in my opinion people's understanding will certainly vary based on their background and knowledge, that is why despite my disagreement with the other poster and you, I try to explain my position repeatedly using various examples so we can at least understand each other to agree to disagree. Having more or less knowledge on a topic does not invalidate a persons ability to think and if I am using knowledge other people don't it is my responsibility to convey that knowledge to hear their opinion. Not necessarily dismiss people because I know some obscure bit of knowledge that they don't.

OMFG. I only bother to read the first 2-3 sentence. Let me address why data is fine here. ANET has the distribution of classes, win rates of each class, and maybe some damage output data, and this data is conditioned on each division. With this data, is it informative to get some idea which class is over performing.

Also, you mentioned you are describing a strategic environment in order to initiate discussion on balance issues, I am simply saying that I dont think game theory is well-suited for this exercise -- its hard to define the strategy space, the utility function etc.

And please, we are not talking about causality here. We are talking about balance... Also, I give an simple example of why model does not matter to make causal inferences. I do a experiment, with a baseline group and a treatment group. I just change one thing between the two groups but I have no idea why I do that modification (this means I have no model, I just do it for fun), but because of random allocation of subjects to these two groups, I can claim I have a causal effect, regardless of I have a model in mind or not...

Also, I know some network theory to understand your traffic example... You and that guy have the same issue. Try to use overly complicated explanation to explain stuff, try to be smart maybe, but a) we dont care in this forum and b) in a formal academic scenario you will get destroyed -- researchers write papers trying to be as simple and as precise as possible, exactly the opposite of what you two are doing. Moreover, its funny that linking hours long lectures trying to backup argument is really really...

Edited June 7, 2021 by Crozame.4098

JusticeRetroHunter.7684 · June 7, 2021

6 hours ago, Crozame.4098 said:

ok, I think Chaos theory is science, but the way you apply it to GW2's balance does not make sense.

So now: tell me how to realistically make the game more enjoyable by not balancing the numbers? IN other words, apply concretely how the chaos theory can balance. Note: concrete.

Alright, see now that I know where your head is at here, I can give you an answer that should clear up misconceptions about what Chaos theory is.

First and foremost is that chaos theory is not science. It is a section of mathematics that is used in many fields and it deals with the inherent randomness of systems. This inherent randomness is caused by the inability to analyze systems with infinite precision.

Tenets

1) The first and most important tenet about chaos, is that all real systems, including completely deterministic ones are inherently random. The reason is again, due to the lack of being able to analyze something to infinite accuracy.

2) The 2nd tenet about chaos, is that, because of the inherent randomness in deterministic systems, any infinitesimally small difference between objects in these systems amplify as the system evolves through time. That means that, even if you know the initial position of some object XYZ to about 6 decimal places, then if you try to predict where that object will be in the near future, you will be completely wrong, and off by a significant margin.

In other words, the ability to predict anything, is based on the knowledge you have over the system... The information you don't know is what's called "Hidden variables." The amount of hidden variables you have available to you about the system, the more accurately you can predict it's behavior. So what happens when the system is highly complex, with many many components where it practically impossible to measure or quantify? Well it means that the system can only be modeled to some approximate notion of it, and can only be predicted to some approximate sense. The nature of this amplfication is exponential, and it's the key reason why we simply can not predict the weather to anything more then 7 days...Prediction becomes impossible because of the sheer number of hidden variables : keeping track of air molecule trajectories.

Hidden Variables

The best way to explain hidden variables is the following example. Let's say you look at some object and want to know what state it is in. You know that it is made out of some number of subcomponents. You look at those subcomponents, and you realize that those are themselves made of even smaller subcomponents. On the scale of the object, all these sub components are called hidden variables... these variables have to be "accounted for" in order to make a more precise measurement...So for example, if your object had 4 subcomponents, and those subcomponent have 4 subcomponents, and those components have 4 subcomponents...then all of a sudden you have about 250 hidden variables that you need to account for in order to understand with 100% accuracy the state of this object.

Guild Wars 2 Example One

Suppose now you are a game developer and you pose a simple question. "Will a single change to skill A balance the entire game?" because of chaos, to answer this question means you need to know all information about not only the skill itself...but all the information of it's hidden variables. Because Skill A exists in a system with thousands and thousands of other skills, each with their own subcomponents, you have a near infinite number of hidden variables to account for, and so knowing whether a change you make to this skill is going to balance the game is at best a very very terrible and inaccurate approximation.

Guild Wars 2 Ape Brain Example Two

Suppose you are a guild wars 2 forum monke, and you make the following proposals : "Nerf Burning down to 3 seconds from 4 seconds on Skill A, and buff the healing from 1000 to 2000 on Skill B." And you want to ask whether this operation balances the game or gives diversity into the game. Given the information of everything said above, you can ask HOW accurately does this proposal reflect the change you desire on the balance of the system, and then you can also ask how would you model that change on the system. If your model is super basic, then your proposal is equivalent to being garbage and not accurate in any meaningful sense.

Equilibrium

Everything else about equilibrium and other cool stuff like fractals, is just a consequence of the two tenets listed above. To find out whether a system is in equilibrium, requires knowledge about the system. The math in order to figure that out is as Kolzar outlines it...you either map EVERY hidden variable to a coordinate in space (called phase space), and then you analyze the change in the system as it evolves through time via some function in the form of a vector that describes it's trajectory through the space. Or, If you do not know what the hidden variables are, then you treat those hidden variables as noise in the system, and statistically, this noise cancels out the larger the system is. This is what it means to model a system in a stochastic process. This modeling just comes from a notion that randomness is similar to white noise, in which noise is in essence uniform across a system, so if you were to map out white noise as vectors, these vectors overall cancel each other out, and you get a more and more accurate measurement of the systems behavior in the form of statistical analysis.

Conclusion

So in conclusion, Chaos Theory is not a science. It is rather just a mathematical truth of mathematics. This truth has many interesting repercussions on the analysis of systems, and creating models of those systems...The sheer amount of applications this theory pops up in is staggering...and really it's no surprise when you realize that all systems unequivocally exhibit chaos.

Edited June 8, 2021 by JusticeRetroHunter.7684

Kolzar.9567 · June 8, 2021

1 hour ago, Crozame.4098 said:

OMFG. I only bother to read the first 2-3 sentence. Let me address why data is fine here. ANET has the distribution of classes, win rates of each class, and maybe some damage output data, and this data is conditioned on each division. With this data, is it informative to get some idea which class is over performing.

Also, you mentioned you are describing a strategic environment in order to initiate discussion on balance issues, I am simply saying that I dont think game theory is well-suited for this exercise -- its hard to define the strategy space, the utility function etc.

And please, we are not talking about causality here. We are talking about balance... Also, I give an simple example of why model does not matter to make causal inferences. I do a experiment, with a baseline group and a treatment group. I just change one thing between the two groups but I have no idea why I do that modification (this means I have no model, I just do it for fun), but because of random allocation of subjects to these two groups, I can claim I have a causal effect, regardless of I have a model in mind or not...

Also, I know some network theory to understand your traffic example... You and that guy have the same issue. Try to use overly complicated explanation to explain stuff, try to be smart maybe, but a) we dont care in this forum and b) in a formal academic scenario you will get destroyed -- researchers write papers trying to be as simple and as precise as possible, exactly the opposite of what you two are doing. Moreover, its funny that linking hours long lectures trying to backup argument is really really...

Getting data on why something is overperforming will not tell you why it is so. Say some build of rev overrepresented, is it because its counters are not viable or is it because something broken with the class. Just randomly nerfing random aspects will lead you nowhere. Regarding economic modelling you don't need a proper utility function to be able to define an environment, I do have the action space. In my very first post I thought was crystal clear about it, finitely many classes finitely many amulets finitely many sigils and finitely many builds rather simple action space you can disagree and say I am omitting some stuff and I will be happy to elaborate why I think this is sufficient to capture a notion of diversity. I am not making any prediction, and once again, once you describe the environment you can describe preferences including whatever limitation you wish to have with whatever solution concept you may want to have. You might say my description is missing some important aspects, but I am simply not understanding why you are claiming its invalid basing off of assumptions I have not made. As far as succinctness goes, when writing a research paper we do know who our intended audience is and I can assume some level of knowledge which allows me to use precise mathematical descriptions. If I am writing in a public forum where I have an opinion and I ask for input from all players I need to try to give examples and explanations that people can relate to and if I think they are wrong I try to illustrate it further again using their examples or things they can relate to. Simply telling you you are wrong because your knowledge of models of stochastic games and bounded rationality is limited does not accomplish anything. I have tried to highlight to you that Nash equilibrium, has its shortcomings such as rationality, but you have models that does not make such assumptions, but just require more mathematical maturity. I gave you examples on misspecified learning, time inconsistency and k-level rationality just because again I think those are at least currently fairly popular. (For example we can define an equilibrium when people don't understand the environment they are in, and they are distinclty misspecified.) Surely you can just start from some subjective model, and preferences defined over them, identify a proper charge and do some analysis. Again just to hammer this in, utility functions are convenient tool but if the underlying preferences do not satisfy some assumptions you can certainly do some analysis without them, you learn them in undergraduate because the only notion of maximization that a typical student knows is through calculus, if you had gone beyond simultaneous games to even very basic extensive form games you should have learned that the set of histories can be defined independently from the players' preferences over them. And even then your model will have some refutable assumptions of behavior, which is completely unnecessary just to define a notion of diversity. And finally, I am not linking any lectures and I am certainly not destroyed in formal academic scenarios. I certainly seem to fail to address a fully heterogenous group over a forum because I have no idea of the background of the people I am talking to and I am being very very long winded.

Edited June 8, 2021 by Kolzar.9567

Crozame.4098 · June 8, 2021

7 hours ago, JusticeRetroHunter.7684 said:

Alright, see now that I know where your head is at here, I can give you an answer that should clear up misconceptions about what Chaos theory is.

First and foremost is that chaos theory is not science. It is a section of mathematics that is used in many fields and it deals with the inherent randomness of systems. This inherent randomness is caused by the inability to analyze systems with infinite precision.

Tenets

1) The first and most important tenet about chaos, is that all real systems, including completely deterministic ones are inherently random. The reason is again, due to the lack of being able to analyze something to infinite accuracy.

2) The 2nd tenet about chaos, is that, because of the inherent randomness in deterministic systems, any infinitesimally small difference between objects in these systems amplify as the system evolves through time. That means that, even if you know the initial position of some object XYZ to about 6 decimal places, then if you try to predict where that object will be in the near future, you will be completely wrong, and off by a significant margin.

In other words, the ability to predict anything, is based on the knowledge you have over the system... The information you don't know is what's called "Hidden variables." The amount of hidden variables you have available to you about the system, the more accurately you can predict it's behavior. So what happens when the system is highly complex, with many many components where it practically impossible to measure or quantify? Well it means that the system can only be modeled to some approximate notion of it, and can only be predicted to some approximate sense. The nature of this amplfication is exponential, and it's the key reason why we simply can not predict the weather to anything more then 7 days...Prediction becomes impossible because of the sheer number of hidden variables : keeping track of air molecule trajectories.

Hidden Variables

The best way to explain hidden variables is the following example. Let's say you look at some object and want to know what state it is in. You know that it is made out of some number of subcomponents. You look at those subcomponents, and you realize that those are themselves made of even smaller subcomponents. On the scale of the object, all these sub components are called hidden variables... these variables have to be "accounted for" in order to make a more precise measurement...So for example, if your object had 4 subcomponents, and those subcomponent have 4 subcomponents, and those components have 4 subcomponents...then all of a sudden you have about 250 hidden variables that you need to account for in order to understand with 100% accuracy the state of this object.

Guild Wars 2 Example One

Suppose now you are a game developer and you pose a simple question. "Will a single change to skill A balance the entire game?" because of chaos, to answer this question means you need to know all information about not only the skill itself...but all the information of it's hidden variables. Because Skill A exists in a system with thousands and thousands of other skills, each with their own subcomponents, you have a near infinite number of hidden variables to account for, and so knowing whether a change you make to this skill is going to balance the game is at best a very very terrible and inaccurate approximation.

Guild Wars 2 Ape Brain Example Two

Suppose you are a guild wars 2 forum monke, and you make the following proposals : "Nerf Burning down to 3 seconds from 4 seconds on Skill A, and buff the healing from 1000 to 2000 on Skill B." And you want to ask whether this operation balances the game or gives diversity into the game. Given the information of everything said above, you can ask HOW accurately does this proposal reflect the change you desire on the balance of the system, and then you can also ask how would you model that change on the system. If your model is super basic, then your proposal is equivalent to being garbage and not accurate in any meaningful sense.

Equilibrium

Everything else about equilibrium and other cool stuff like fractals, is just a consequence of the two tenets listed above. To find out whether a system is in equilibrium, requires knowledge about the system. The math in order to figure that out is as Kolzar outlines it...you either map EVERY hidden variable to a coordinate in space (called phase space), and then you analyze the change in the system as it evolves through time via some function in the form of a vector that describes it's trajectory through the space. Or, If you do not know what the hidden variables are, then you treat those hidden variables as noise in the system, and statistically, this noise cancels out the larger the system is. This is what it means to model a system in a stochastic process. This modeling just comes from a notion that randomness is similar to white noise, in which noise is in essence uniform across a system, so if you were to map out white noise as vectors, these vectors overall cancel each other out, and you get a more and more accurate measurement of the systems behavior in the form of statistical analysis.

Conclusion

So in conclusion, Chaos Theory is not a science. It is rather just a mathematical truth of mathematics. This truth has many interesting repercussions on the analysis of systems, and creating models of those systems...The sheer amount of applications this theory pops up in is staggering...and really it's no surprise when you realize that all systems unequivocally exhibit chaos.

1) Fine, math is not science. also, Statistically, or econometrically, unknown variable should not be treated as noise. You understand the term omitted variable bias?

2) Suppose true shot is hitting 20k. So we should not tune down the numbers, but instead should increase the "complexity" of the game, by adding more absurd numbers?

3) I remember in someone said in 2 sentences, without invoking any of your nonsense, that if a certain class is OP, then should not nerf, rather should make a counter to that... This is also undesirable. Because this will make the game even more comp based, not skill based.

4) I mean seriously, you still did not suggest how to balance. It like there is a problem, we propose solutions to the problem, but you keep saying we are not right, you know the correct solution, but not saying it...

Crozame.4098 · June 8, 2021

7 hours ago, Kolzar.9567 said:

Getting data on why something is overperforming will not tell you why it is so. Say some build of rev overrepresented, is it because its counters are not viable or is it because something broken with the class. Just randomly nerfing random aspects will lead you nowhere. Regarding economic modelling you don't need a proper utility function to be able to define an environment, I do have the action space. In my very first post I thought was crystal clear about it, finitely many classes finitely many amulets finitely many sigils and finitely many builds rather simple action space you can disagree and say I am omitting some stuff and I will be happy to elaborate why I think this is sufficient to capture a notion of diversity. I am not making any prediction, and once again, once you describe the environment you can describe preferences including whatever limitation you wish to have with whatever solution concept you may want to have. You might say my description is missing some important aspects, but I am simply not understanding why you are claiming its invalid basing off of assumptions I have not made. As far as succinctness goes, when writing a research paper we do know who our intended audience is and I can assume some level of knowledge which allows me to use precise mathematical descriptions. If I am writing in a public forum where I have an opinion and I ask for input from all players I need to try to give examples and explanations that people can relate to and if I think they are wrong I try to illustrate it further again using their examples or things they can relate to. Simply telling you you are wrong because your knowledge of models of stochastic games and bounded rationality is limited does not accomplish anything. I have tried to highlight to you that Nash equilibrium, has its shortcomings such as rationality, but you have models that does not make such assumptions, but just require more mathematical maturity. I gave you examples on misspecified learning, time inconsistency and k-level rationality just because again I think those are at least currently fairly popular. (For example we can define an equilibrium when people don't understand the environment they are in, and they are distinclty misspecified.) Surely you can just start from some subjective model, and preferences defined over them, identify a proper charge and do some analysis. Again just to hammer this in, utility functions are convenient tool but if the underlying preferences do not satisfy some assumptions you can certainly do some analysis without them, you learn them in undergraduate because the only notion of maximization that a typical student knows is through calculus, if you had gone beyond simultaneous games to even very basic extensive form games you should have learned that the set of histories can be defined independently from the players' preferences over them. And even then your model will have some refutable assumptions of behavior, which is completely unnecessary just to define a notion of diversity. And finally, I am not linking any lectures and I am certainly not destroyed in formal academic scenarios. I certainly seem to fail to address a fully heterogenous group over a forum because I have no idea of the background of the people I am talking to and I am being very very long winded.

1) I said data is informative? I did not say it will provide the definite answer? The devs can then investigate further by watching vids or trying out themselves. Why you are assuming I meant randomly nerfing something?

2) Because it is not possible to define the strategy space during game play.

3) Because you cannot assume players are rational or even optimising something, this is a game, without a competitive scheme, no one cares that much. This is not true for using economics modelling to study real life interactions, because 1) it matters a lot to people, 2) there are firms, institutions and others that are close to rational and 3) there are markets and the price of the power is huge -- you know the market efficient hypothesis?

4) I said experts right in simple and short terms, they do not explain stuff by posting hour long vids.

5) You know for level-k reasoning, you need to have some knowledge about the distribution of types to conduct meaningful analyses?

6) I am confused. Utility functions are required also in extensive form game: the definition of utility function in game theory is mapping from actions to payoffs. Tell me, which branch of game theory does not require utility functions? You trying to be smart here, but please understand the basics first.

7) I am also confused about your definition of history... Loosely speaking a history is just a set of moves taken by the players -- yes it is unrelated to preferences. But history ends at a terminal node. And you should know that terminal nodes is followed by utilities. And utilities already taken into account players preferences.... You trying to be smart here, but please understand the basics first.

JusticeRetroHunter.7684 · June 8, 2021

5 hours ago, Crozame.4098 said:

4) I mean seriously, you still did not suggest how to balance. It like there is a problem, we propose solutions to the problem, but you keep saying we are not right, you know the correct solution, but not saying it...

Here's the real issue...forget the solution. You don't even know the parameters of the problem or how to model it. If you learned anything about what I just said above, you'd realize that the problem itself is much larger in scope.

Quote

Statistically, or econometrically, unknown variable should not be treated as noise

This is completely false. You do this because you can't actually GET the unknown variables...it requires measurement you can't do in practice...so you just have to trust that your hidden variables will cancel out given a large enough statistical sampling (or until you get a better measurement apparatus)

Quote

2) Suppose true shot is hitting 20k. So we should not tune down the numbers, but instead should increase the "complexity" of the game, by adding more absurd numbers?

This here is an ape level analysis of the problem. Because this skill hitting for 20k, is dependent on the existence of how well other classes are able to defend themselves. Even looking at the damage equation on the wiki points to the fact that this number is not even a static quantity. it may hit 20k against your super glass naked staff ele. but it hits for 4k, if at all, against a dodge bot Strength Warrior.

About complexity, you are just making ridiculous assumptions about what you think the solution is to a problem you don't even fully understand in any meaningful scope. Increasing complexity is not "adding more absurd numbers."

If you actually did look into anything on complexity theory, you'd realize it is the same THING as Network theory that you said you studied before. I think if you actually took the time to listen and learn from other people and to stop lying, you wouldn't even be here making these kinds of really thoughtless comments.

Edited June 8, 2021 by JusticeRetroHunter.7684

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