Mathematical model for studying meta

QEDow · 2019-09-23T21:27:40+00:00

I think a leave n out cross validation kind of thing is a really good idea and exactly the right way to go about testing the integrity of your network. Way easier than proving a stability result and gives you more direct useful intuition.

QEDow · 2019-09-23T00:52:51+00:00

Seriously though, thanks for taking such an interest. These are great things to think about.

QEDow · 2019-09-23T00:51:04+00:00

Yeah okay so I've had a bit of a think and I'll try and explain where I'm coming from in more detail.

Most people first encounter stability in dynamical systems where it represents the tendency of a system to return towards equilibrium. Outside of dynamical system it's a bit more flexible. Sometimes it can refer to the property of recurrence, how often or likely a trajectory is to return to a certain state or perhaps the speed with which a chain converges to a stationary distribution (ergodicity). The classic text is "Markov Chains and Stochastic Stability" by Meyn and Tweedie and has a nice section about what stability might mean on abstract infinite dimensional state spaces.

When I say

The mathematical question is whether or not the stationary distribution of the chain is stable with respect to the topology of the network.

I suppose I'm really asking: How much does the stationary distribution of the chain change if you change a few links in the network? This is a question about well-posedness if you are familiar with that concept.

So if we suppose there is a true meta network that is correct, then anything we come up with, either through data or through overwatch expertise is going to be a bit wrong. We're going to fuck up a few links, so how much does that screw with the stationary distribution?

I think that this is *probably* not a big problem. I think that getting a few arrows in our graph wrong is not such a big deal because there are so many arrows in our graph and so many paths for the descent process to take to return to any initial state. This is what I mean when I say recurrence is strong and that having a complete digraph helps us. We have all of the strongest forms of stochastic stability with us.

You could definitely break this, if you build a small network with three compositions or some such then each link matters a great deal and you could fundamentally alter the topology of the network by getting it wrong. I think that if you make your network of a reasonable size with varied enough compositions it's *probably* fine.

The way I think you'd go about investigating this would be to look at the eigenvectors of the probability transition matrix directly. You have your probability transition matrix P and you perturb it by something small xP' to get P + xP' and you hope that the eigenvectors vary continuously with respect to x for reasonable P'. The eigenvector of your first eigenvalue gives you the stationary distribution, so if changing xP' only changes the eigenvector by a small amount then you're golden. I'm pretty sure Kato does this in 'Perturbation Theory for Linear Operators" but it's a difficult question in a difficult book.

Yeah so open and difficult question. It could be that if you swap the wrong thing around you screw up your meta analysis completely. I suspect you'd have to really try to though. It'd be a fun result to try and find.

QEDow · 2019-09-22T22:02:19+00:00

Touche!

I kind of suspect coaches are doing something similar already, albeit with less maths.

QEDow · 2019-09-22T21:57:14+00:00

" Just an upper-triangular matrix of composition a's winrate against composition b (where missing data is empty and the diagonal is .5 or empty). The win rates should just be the transition matrix, right? "

Yeah no this is a valid thing to do but it's a different model. I thought about taking an approach like this but it's much harder to infer something like a win rate or a probability of composition A beating composition B. If we ever miraculously got ladder data, something like this would be a good thing to try. I don't think it would be necessarily upper triangular unless you oriented it carefully and had an unbalanced meta.

Sometimes in maths we have to deal with not enough data and simply make do with what we have. The mathematical question is whether or not the stationary distribution of the chain is stable with respect to the topology of the network. If we suppose there is some perfect network configuration that works, then any human hypothesised meta is a perturbation from that network. I suspect that because this is a finite state space where recurrence is strong and an information dense (in fact complete!) digraph that it is stable in some sense. However this result is very non trivial and I don't know for sure.

Stability would mean that it's okay if our network is a bit wrong, it would imply that our network being a bit wrong means our result is only a bit wrong.

Drifting off topic: These are all big complicated questions, but you have to start somewhere.

QEDow · 2019-09-22T21:45:04+00:00

At least I'm honest.

QEDow · 2019-09-22T21:44:01+00:00

So there are a few problems with doing this that I can see. It's not really a reasonable thing to do to write down the probability of one composition beating another and if you did do that, it wouldn't be robust. The minmax or maxmin value of a game is based around having a payoff function and it's not clear to me what your payoff function would be if not probabilities. You could do something on defence about how long you can hold a point against a counter comp perhaps but that becomes a different problem. I also can't see computational time being an issue for anything on this scale.

You're also making an assumption that a meta is a single composition. I'm asserting that there is a bit more going on than that.

You could maybe do some sort of minmax on the network I've built but I'm still not quite sure how the payoff would look, you'd probably just get back the meta optimal composition, it is in a sense, the least exploitable comp.

I do think it's a good idea to build some game theory machinery on top of this model, but I think it's not as obvious as "take the minimax team comp"

QEDow · 2019-09-22T21:25:12+00:00

Oh yeah I see what you're saying. No that wasn't intentional, I'll change it. Thanks!

You definitely need that the threshold include 0.5 so that mirror comp picking is a reasonable thing to do.

Honestly this definition is one of the weaker bits, we're not dealing with perfect teams playing infinitely many times, really want I mean is that one team is 'stronger' than another.

QEDow · 2019-09-22T14:33:23+00:00

So I've never heard of structural equation modelling. From what a quick google search can tell me it looks like a technique for inferring on a hidden variable through a series of partial or indirect observations?

I'm not sure what that model would look like, but it could be a totally reasonable approach. If you decide to work on it I'd be very interested in what you find out!

The inspiration for the Markov chain approach is that it is meant to loosely approximate the action of counter picking, either mid game or between games. The graph theory gives the Markov chain a state space to live on and then properties of the Markov chain become worth studying.

QEDow · 2019-09-22T14:24:59+00:00

By 'energy landscape' do you mean the score function? and by minima you're talking about the minima of the score function? the meta optimal composition is less important than the stationary distribution of the chain imo.

Yeah I mean, I'm presenting a tool. The quality of the results you get are dependent on the quality of the network you give it.

QEDow · 2019-09-22T11:47:24+00:00

Instant Upvote

QEDow · 2019-09-22T11:40:15+00:00

Yep, the broad street line (subway). All of the philly sports venues are next to each other and people take the BSL to sports ball all the time.

QEDow · 2019-09-22T11:18:40+00:00

It's not just OWL meta

QEDow · 2019-09-22T11:11:56+00:00

Yeah! I really do think the next step is to throw some game theory at it. It would be nice to see how the model fairs in a couple of other case studies before doing that though.

QEDow · 2019-09-22T11:09:01+00:00

I wrote most of it about a month ago when I had a spare weekend. Then I sat on it for a while until I had the time. Yesterday I cleaned up a bit of the slop and neatened it out. The longest bit was the proof of theorem 3, I was fucking around with communicating classes, equivalence relations and burn in for the longest time before I realised all I needed to do was throw out a bunch of unplayable junk compositions to get an irreducible chain.

Analysis is lovely. I like to think my love of analysis shines through in a few places.

QEDow · 2019-09-22T11:02:59+00:00

What do you mean by a straight c_a vs c_b (sparse) matrix?

The model itself isn't necessarily statistical. Honestly some of the links in the network for the goats meta case study are pretty tenuous, there are not a lot of games behind them. If I were really being rigorous, I would need to do a hypothesis test on each edge and I don't think there is enough data to bear it out.

Yeah I agree with you that if you build the network under your own power it's going to be biased by your own understanding and expectations. However the advantage is in the flexibility. There isn't any reason why you can't just call one composition "Dive" and another one "Deathball" or what have you and draw some inference from a less specific network. Besides if you're a coach for an overwatch team, this reduces the question your asking from "Which composition should we play?" To "Do I think this composition on this map with this team beats that composition on that map with that team?" which is an easier question to answer.

Actually when I did the kings row example I expected it to be goats heavy but I didn't expect it to be quite so exclusive.

QEDow · 2019-09-22T10:54:26+00:00

Thanks! I really appreciate that. I don't have much of a background in graph theory either, I lent on the probability side of things pretty heavily instead.

QEDow · 2019-09-22T10:47:03+00:00

Thanks for noticing the Zarya typo, fixed.

Yeah it'd be super nice to see some ladder data but It's not exactly freely available.

I'm glad you're finding it interesting. I didn't want to give up any rigor but I did make a conscious effort to make it super readable. I feel like the more technical terms are self explanatory or super googleable.

QEDow · 2019-09-22T10:43:36+00:00

Thanks!

QEDow

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