Kolzar.9567

Your only condi clear is jaunt?

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

I think you are not paying attention to the relevant spaces of these mappings. I gather you are more familiar with functions instead of correspondences so I will try to use that, but in the end they hopefully the relation to my earlier posts will be clear. And again all I am suggesting are different notions of diversity, so that when we talk about diversity of meta we have the same thing in mind, as I still think complexity is not appropriate. A function as you might recall, has a domain and a range, and has to take every single element in the domain and has to map it into a single element in the range. Now, lets start from our domain as the set of builds in gw2 (it really doesn't matter how you represent them be it vectors or other things, at the very least you can literally just name them). Say our very simplified space is {scourge, herald, holo, weaver}. If I think of a function on this space it takes every element and spits out a single element. So if we somehow think that this mapping (which we did not make further assumptions on) is representative of how meta evolves (again including all my limitations of the player base) as a counter for a build, if I think of this as a function it restricts me to a single build, although I might be thinking both holo and weaver counters scourge. Furthermore since I am thinking about some collection of builds that are in the meta, and their counters even the domain doesn't do me much good, since the current meta could be sourge and holo, not just singleton objects This part I can fix partially by thinking what counters scourge, holo, individually, and just taking unions, but it doesn't help with the fact that I may have more than one counter. So clearly we need to either change the space or get something more broad then a function. Earlier I directly went with the latter, but this time lets go the former way, however keep in mind I am still trying to describe the same situation and get the same notion. Now I consider the set of all non-empty subsets of the set {scourge, herald, holo, weaver}, this object as you might recall has 2^4 -1 elements, and looks like {{scourge,holo},{scourge,herald,holo},{weaver,holo}{weaver} .....}, all the single, double, triple element subsets and the full set itself. Now a function on this space would take a subset say {scourge,herald, holo} and map into say {weaver, holo}. However we gained some freedom, since the singletons are here, so you can say the counter to a {scourge} meta is {holo, weaver}. Now a fixed point on this function on this space of subsets, is going to be an element of this space, a set. So whatever notion you can have with regards to subsets you can use here. I suggested the size of this set, hence its cardinality. It really has nothing to do with the mapping, but basic properties of the elements. To make an extreme example, suppose we were mapping apples to apples, we could think about the redness or freshness of a fixed point, since it is going to be an apple. Now instead of just builds, we could have started from builds and their frequency in population, which would be probability distribution over the set of builds. An example would be {Scourge 1/3, holo 2/3, herald 0, weaver 0} that is currently 1/3 of the population is playing scourge, 2/3s are playing holo, noone plays herald or weaver. Again there is a space of such distributions, usually denoted of the simplex over this set (not to be confused with the simplex algorithm), and usually denoted with delta. So our space would be Delta{Scourge, Holo, Herald, Weaver}. Again we can define a function here, this time from this space of distributions to space of distributions, and again a fixed point will have whatever we can define on an element. Now since these are probability distributions candidates could be standard deviation, mean, kurtosis, etc, or again just the size of the support of the distribution that is the fixed point. The advantage of such a formulation might be a better representation of reality, the disadvantage is that you are dealing with a more complicated object. With that in mind to reiterate why complexity is odd here. Nash equilibrium, or loosely speaking any equilibrium/stability notion can usually be described as a fixed point. The complexity of calculating very well behaved fixed points such as even Nash equilibria is usually bordering on NP (there are lower complexity ones such as contraction mappings, but those are rather special). And Nash equilibrium is a beast, it assumes way too much about what people know and how people think that is why it is so well behaved. If you think a fixed point of how fotm builds change/evolve is an accurate description of a stable meta you are going to run into the same issues and more. (Again mine is a mere suggestion, you can certainly disagree with all or parts of the descriptions in paragraphs 2/3) The moment you start relaxing some of these assumptions so you get weaker solution concepts than Nash (and you really should given that you don''t know why people play the game), the complexity as defined in the standard way increases, just because now you need to also consider how people think, what people know about the strategic interaction they are in, each other etc.

I am not saying complexity is ill-defined or it doesn't exist. I am just saying since this is a fixed point on some correspondence, the complexity class is not representative of how long people will actually take to reach a certain point. If you google equilibria/fixed points and complexity classes you will see that even simple settings we encounter and even solve in real life are just too hard by this classification, because algorithms usually can't take the mental shortcuts/fallacies we have.

Actually, I am not attempting to classify the game in terms of any complexity as I am deliberately trying to avoid making assumptions about why and how people play the game. All I am saying is regardless what you assume about why people play the game, the very nature of how a meta evolves, a set of builds changing into another set of builds resulting from some sort of attempted optimization by players is very complex by classifications such as NP, P etc, just because it is some sort of fixed point on a correspondence. Such complexity is present in very simple examples as well, such as repeated plays of RPS just because of the nature of the problem (and how it interacts with the formal definition of complexity) and the usual notion of more complex=hard for people too and it will take soo much time is lost. At least in my opinion, the complexity class of say RPS, or GW2 is not representative of either the difficulty of the game or the diversity of actions/builds that are viable in the game. (Again I am not saying the game itself is difficult or not, just using complexity as a measure at least in my opinion is not very appropriate) And since I disagreed with you on this part, I tried to offer some alternatives of a notion of diversity which I think are better suited, again these may or may not be appropriate but the whole point was to at least even find a notion of diversity that people can to some extent agree on, so that you can discuss how to increase/decrease it.

That was what I was trying to explain, but there is a little more to it. It is a process defined on sets of builds. So unlike processes you would encounter in physical phenomena (say currently my cup of tea is at 30 degrees Celsius at 10 seconds from now on, it will be something slightly below, with some randomness in it and I have neat formula to calculate how I expect it to evolve), it is prone to exhibit some problematic behavior (also it has some good behavior because people are doing what is in their best interest given their understanding of the game, but that is a separate issue) that makes complexity of the problem overinflated and at least in my opinion inappropriate to use. (the very crude extension to the tea temperature would be that my tea would simultaneously be 25 and 10 degrees at 10 seconds, since it need not be a single value) When you think about a physical evolution, there is at least a very loose notion of "time derivative", how your process evolves with respect to time. Now, as you might remember, a notion of derivative requires continuity. Suppose you are currently at x, and expect to evolve to y, if you instead were a tiny bit from x, you would expect to be near y. However, when you are thinking about set valued objects, the notion of continuity also needs to have sets shrinking expanding. Say currently everyone is playing prot holo for some reason, and there is only one counter (I am not pushing any reasons why there is only one, that people can argue separately), so the next evolution in the meta would be that build. But suppose we changed something tiny on the prot holo say we removed one amulet or sigil, now there are 5 counters (again no assumptions on why the player base plays those as counters), which may or may not include the original counter we had in mind. This rapid, discontinuous expanding of the set is a failure of a notion called hemi-continuity. This failure automatically makes your problem very complex regardless of the rest of the real situation, and whatever you assume about player behavior, which makes using complexity not suitable in my opinion. And this kind of complexity usually does not pose a problem for human beings for simple enough tasks, or even complicated ones given enough practice, even though they may have a plethora of cognitive limitations that you would usually assume away. I tried to give examples before, R,P,S is a very complex game by this notion, yet kids play it, supply equals demand is a very simple concept , it is a well known NP hard problem yet markets even with bad business decisions seem to operate and even have some sort of stability. You can try to analyze these situations with a solution concept you have in mind where you would impose some assumptions on behavior(what players want, what they know how they react etc) that should in principle make your life easier yet doesn't get around the inflated complexity.

This is literally the way to have a meaningful discussion in a sensible framework. You might already disagree on the description of the environment and in fact that was the reason of my first post, I disagreed with the notion of diversity that the other poster posed, we kept respectfully disagreeing and they seem to agree with some of my choices. I am making at least in my opinion some minimal assumptions of my description, the other poster does another set of them. I tried to explain to them why some of those assumptions they are making would be invalid (people behaving randomly does not generate enough continuity to think of it as a mechanical equation), they came up with their own arguments. I tried to keep the level low so that other users can also understand what I was saying and agree/disagree, maybe I missed something in my description, maybe I didn't. Once a reasonable framework is established via discussion, then you can impose some behavioral(i mean this very loosely, essentially assumptions on the black box you left for preferences/knowledge/rationality, anything that will govern the path of play) assumptions, you can say they come from economic theory, you can say they come from some other theory, people lay out their assumptions you can try to agree or refute, using observational data/logic/math whatever is at your disposal. If your behavioral assumptions are sufficiently justified, then you can try to generate solution concept, which again you can justify/refute and check using data/analysis whatever strikes your fancy. So it is fundamental to have meaningful framework to facilitate the rest of the discussions, say where did Prot holo came from. This is what you would do for real economic concepts as well, although I have given this example before, say you want to have a notion of inequality, for the time inconsistent retirement problem. Even though your economic agents objectives and knowledge may be unbeknownst to you or they are so complicated that you don't have a good notion of solution. Once you have your notion you can talk about it, try to put structural assumptions on for example form of inconsistency etc, ask other people, you know improve your understanding on the problem at hand. Once again I feel like I am being led to a random gotcha instead of a meaningful discourse if that is so, again please stop asking me questions. I am readily willing to give you that you are right and I am wrong on whatever thing you believe I am wrong in.

For the very last time, look up my post from 4ish hours ago, regarding points you raised 2-6-7) and the following example. I am not trying to solve a game, I am defining histories and defining a notion of diversity on these histories, no solution concept, no need for utilities or preferences or whatever, just a description of the environment. As an outside observer, monkeys do whatever monkey like, I am not trying to make any prediction. I simply say look at the history, if it is repeating look at the support of the actions in the history. I don't care where that history comes from. The references to notions of bounded rationality and how to evaluate outcomes in such environments only came up because you brought them up and I elaborated, with references at your request which evidently was my mistake. If you are unsatisfied with this answer, you can simply assume I am wrong, you are right, and conclude the argument as such, but I would appreciate if you would stop quoting me to ask questions.

Look up rational preferences and rationalizable choices, you will immediately see that it is not your run off the mill VNM expected utility. Although at this point I give up, you are just being argumentative without answering where we needed utility functions to describe the dynamics. At your next step you can proceed to correct my grammar as well. As you alluded regarding the other poster I don't know how to argue with someone like "you".

To describe the environment and the path of play you don't need preferences, to impose a solution concept you do. How do you think people are even thinking about strategic environments where people are not fully rational? Once again what kind of assumptions did you need to describe what was happening on the monkeys and humans playing RPS. You could literally just write down the path of play. If I assumed monkeys or humans preferences over paths of play that are well behaved enough you can define utility representations for them and try to solve the game, I have not attempted to do so. This is a basic modelling concept beyond game theory even any kind of single player stochastic control problem has the same virtues. Suppose I am literally typing paragraphs using random letters, the set of letters I include in each paragraph is going to be a process, regardless of the objective I have and if you see that I start typing the same letters you can say something. Now, you can say I know some languages and I am typing coherent sentences, or not, you do not that bit of information to see whether I have started repeating myself. If you have a full fledged well behaved utility function with an nice accompanying objective probability measure you can say well I will try to use these well behaved solution concepts such as NE, SPNE, SE etc. If you have instead subjective probability measures for example, then you end with preferences with ambiguity, you may have a representation, but it will probably at best be of Epstein-Zin form (which you can actually put into formal models and use it to try to explain various puzzles in finance), at worst could be just an integral of a charge you get out of Savage or Anscombe-Aumann axioms. You want to learn about such constructions (this is the largest branch in non-rational models) you can start from Kreps' book, notes on theory of choice and eventually catch up to the frontier. If you want to think people have misspecified learning, say I don't understand whether candy is good for me or not, but as I keep eating it and I learn you will have a model of misspecified learning, google Berk-Nash equilibrium, you will find wonderful bits of research. If you just want to understand how to define a history space, literally pick any graduate textbook, MWG, Mailath Samuelson, Fudenberg Tirole etc. You will see that they will first define the set of actions, then the set of histories, which is all I have done. In fact you will see they will add different monitoring structures, behavioral types, or limitations of rationality and then compare to the fully rational case. And once again these are textbook examples so they are approximately 10-15 years behind the frontier. Just to highlight that you don't need a valid and known utility function look up rationalizability (it should have a section on MWG). Notably if you look up at any of these books you would see that they will usually start with either a choice correspondence or a preference relationship. Any strategic interaction again, you will need to have description of outcomes to actions, and you have preferences over such outcomes. Just to hammer this in, RPS has actions, and outcomes, who wins who loses, then the players will have preferences over those to have utilities to write it succinctly. You don't need to tell me I get 5 utils from winning 0 from draws or 7 from losing (the preferences need not align by the notion of winning or losing), just to say in this environment the set of allowable actions is RPS, and an outcome will be a tuple of the form (R,P), (R,R) etc and so on and so forth. If I keep repeatedly playing this game the set of histories will be sequences of the form ((R,P),(R,P),(R,P).....). I am not sure what is missing here, that prevents from saying well these guys are playing some stage repeatedly and have some preferences, but they seem to be repeating (R,P). I thought the example with the monkeys and the set of histories above was crystal clear and I don't see an argument about what you didn't understand about that. Did you not understand the description of distribution of actions over time? Did I need to make any assumptions on utility to describe the process and a notion of diversity over there? Clearly you can simply extend it to GW2, instead of having RPS, you just pick a build out of which there are finitely many of. Instead being paired as 2, you get paired as groups of 5, and the random information that is released can be thought of the forums/metabattle whatever. But as I previously mentioned you seem more convinced of your extent of knowledge, rather than trying to look up the things or even entertain the idea that other people might actually know more on certain topics. I am certainly not appreciative of you demanding references a) because you can look them up yourself, b) because again it would just be alienating the people further. I thought the argument I was making was clear enough that I do not need to appeal to any authority including my own, that was the reason of giving so many pedestrian examples. If there is a flaw in the descriptions I have provided you are certainly welcome to point it out directly instead of appealing to some irrelevant half knowledge. As long as you keep it civil I can keep trying to explain it you, but unfortunately I can't understand it for you.

You already seem pretty convinced that your level knowledge is enough, have gone into making assumptions about my knowledge/occupation/expertise so I am not going to even address the argumentative points. 2-6-7) You are just ill informed and you don't seem to have understood what was suggested. In extensive form games/stochastic games you can describe the environment and the set of actions and the resulting history of play without making any assumptions on preferences/knowledge. Extensive form games are not limited to finite sequential action extensive form games only and you can have repeated games with random matching which are still a sort of extensive form game, which i believe is the appropriate for GW2. Just to address the point of a terminal node, you might also want to brush up on your knowledge of repeated games with or without perfect information and see what the set of histories look like. Regardless, even on finite extensive form games, you can draw the game tree just by knowing the set of available actions to the players. In such games a sequence of actions, will lead to an outcome, the players will have preferences over such outcomes. If those preferences are well behaved enough, then you might try to say there is a utility representation. Again, you can define notions on the history of play without making any assumptions on behavior or even a solution concept. That was what I did. To illustrate the point in a simple example that is hopefully accessible to everyone, suppose you collect some number of actual monkeys and some players from this forum, put them each in a cubicle that has three buttons R,P,S. At random intervals in time, you pair up some of those (not necessarily all concurrently) cubicles randomly and have them pick a button have them effectively play RPS over this small internet. The rules for determining the winner are the same as regular RPS, but winner gets a candy, if its a draw they get a juice and the loser listens to 15 seconds of Justin Bieber. If the monkeys or humans don't play you can assign a random action to them, or just assume they lost by taking an additional action F, but lets stick with the first one. You keep doing these randomized matchings and keep recording the frequency of R,P,S that was played, . Congratulations you have defined a random process over the histories. You can even add random timed information release announced into these cubicles, where you say in the last 5 games played R had this frequency, P had this frequency, S had this frequency. So people and monkeys get matched, play a game, maybe get some reward, make their inferences about their own experience or don't and adjust their actions for their next pairing or don't. Now if it at some point after having been randomly matched the players (including the monkeys) the frequency of RPS that was played did not change, you have reached a stable meta (a.k.a a fixed point on the mapping from distribution over actions do distribution over actions) of this jointly controlled process. Now if at that point suppose only R,P is played, then you can say directly well that is not a very diverse meta because only two things are viable, alternatively you can incorporate the standard deviation or some characteristic of the distribution, and say if the deviation is large or not. I suggested both as viable alternatives. Now, to define these notions, what have I assumed about the rationality/knowledge/preferences of the humans or the monkeys? Nothing. Maybe some people prefer juice, maybe some people like Justin Bieber, maybe the monkeys have learned game theory before and know how to bid in a first price auction, I don't care and it is unnecessary. You can vaguely assume they do what they like limited to their tastes/knowledge/preferences/learning along the way and it will buy you a little bit about the fixed point (mind you still no utilities just basic maybe incomplete, intransitive preferences, because variants of theorem of the maximum/zorn's lemma start becoming applicable but it is not even necessary) It does not matter, I don't need it just to describe a strategic environment and a notion on it. On a side remark, utility functions are just ordinal representations of preferences, when these preferences satisfy some assumptions, they are not the primitive to do any economic analysis, they are just convenient. 5) I did not impose a solution concept (like Nash equilibrium) above, you assumed I did. Nash equilibrium does have stringent requirements on rationality and knowledge of the game which limits its applicability. I gave several examples where such requirements are relaxed, just to illustrate the point that if someone wanted to, they might attempt to use a solution concept that is appropriate for whatever kind of relaxation they think is fit. There is an entire subfield of economic theory called decision theory that deals with a large swath of such relaxations. Maybe people are ambiguity averse, maybe people have wrong models in their head that they keep updating, maybe people are loss averse, maybe people are time inconsistent. Most of these models start with a preference relation on acts and you may or may not get a utility representation or even a proper probability measure. If people think their assumptions on these bounded rationality is good enough, they can impose it on the environment and may attempt to make predictions, then you can refute if the assumptions are indeed good enough or ill fitting. 4) Once again I have not posted any videos and once again when writing a paper or giving a talk, you know that referees/editor/seminar audiences will have a basic knowledge of the topic and at the very least will have a PhD (and therefore completed a core sequence) as well as being familiar with the conventions of the profession. This is not the case here, using only technical terms without explanation is just going to alienate everyone.

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.

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.

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.

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.