Hate Speech: Theory Fighter University: Remedial Math

[Synraii]

^ Note he said take the player out of the equation Nirf: this is pure mathematical theory, not how it actually works - its just an underlying guideline. If you take players out of the equation alltogether, it is just a 50:50 and mathamatics can only treat it for what it numerically represents, so it has to assume 50:50 to literally mean 50 one way 50 another.
You run a similair risk reward with any throw/mid character and don't forget Cervy has a pretty high damage output, so I'm struggling to see what you're getting at tbh, would you be able to give me something to clarify as I'm now a tad confused, lol.

 

[Dr. Hates]

In this post you delve quite a bit into game theory. While your comments on SC are generally spot on, there's some errors in how you treat game theory that actually have some important ramifications.

See my response to Signia. I'm not assuming that players will choose any given option half the time, but that players of equal skill will, in a 50/50 scenario, choose properly half the time without regard to what that choice might be.

You're absolutely right that the numbers shift based on the frequency of various options being chosen, but I choose to fudge the specifics here for three reasons:

1. I want to suggest a way of looking at these scenarios that is somewhat concrete without being daunting--avoiding calculus is important to me because I don't think suggesting people do that is palatable.

2. The more we bring in specific elements of player choice, the further we go down the rabbit hole of extreme complexity. What about nonstandard options? What about issues of relative health? Ring position? By assuming it's a wash with players of equal skill (which is perhaps the biggest limb on which I'm going out, so feel free to attack that), we can at least examine something tangentially related to reality.

3. My purpose is to provide a shorthand tool that can prompt some reflection and get people to consider things from a different perspective, not provide the Ultimate Flowchart.

If you want to go ahead and run some more complicated numbers, that'd be awesome and I think people might benefit greatly from it. In particular if, when doing so, you can provide a relatively simple model for others to make similar calculations.

Optimally, Astaroth can mostly bullrush and very occasionally grab. Cervantes will then follow a strategy of usually blocking, and very rarely ducking The damage will then be in Astaroth's favor.

You're not wrong here, which I think thoroughly underscores my point about why it's always in your best interest to do everything possible to force your opponent into a zero-sum mixup scenario. If Cervantes stalwartly refuses to do that, however, we're left to contend with figuring out the relative benefit he receives from the free mixup he gets upon blocking every bullrush. Those will translate into damage, but they'll also expose him to risk, ad infinitum. See why I chose to draw arbitrary lines and simplify? ;)

 

[Nirf]

So here's the real math.
Assume Astaroth has two choices: B (bullrush) and G (grab).
Cervantes has two choices: C (crouch) and G (guard).

Damage outcomes for the payoff matrix:

BG: Asta bullrushes, Cervantes guards, = 0.
BC: Asta bullrushes, Cervantes crouches, = 28 damage.
GG: grab, guard = 36* damage.
GC: grab, crouch = -70 damage (i.e. Asta takes 70 damage).

* this number is actually incorrect to for the same reasons as the larger argument is incorrect: the grabs and breaks will not be 50/50. I use it out of laziness, in reality it's LOWER.

Asta randomly decides what to do, he bullrushes with probability p.
So he grabs with probability 1-p.
Cervantes crouches with probability q.
So he guards with probability 1-q.

The expected payoff for Astaroth is:

E = BG * p * (1-q) + BC * p * q + GG * (1-p) *(1-q) + GC (1-p) * q

So now what? Well, Asta can change p to try to maximize E, and Cervantes can change q to minimize E. The equilibrium (yes, Nash) occurs when for some value of p and q, they are at a maximum and minimum at the same time. At a max or a min, we know the slope of the curve is zero with respect to the optimizing variable, so we set:

dE/dp = 0 = BG(1-q) + BC * q - GG * (1-q) - GC * q

dE/dq = 0 = -BG * p + BC * p - GG * (1-p) + GC * (1-p)

We're lucky here in that we get to solve for every variable separately. Mathematically there are some things that need to be verified to make sure this is the solution, but yall can trust me, I'm pretty sure it's right.

q = -(BG-GG)/ (BC + GG - BG - GC)

p = -(GC -GG)/ (BC + GG - BG - GC)

This is all completely general so far btw, and can be applied to MANY situations in the game. Plugging in, we get:

q = -(0 - 36) / (28 + 36 - 0 - (-70) = 36 / 134
p = -(-70-36) / (...) = 106 / 134.

These results actually make intuitive sense. Astaroth's p is far over 50%, that is he will bullrush a lot more because it's safe. Consequently, it makes sence for Cervantes to have a low q; with Astaroth bullrushing so often, Cervantes better not duck too much. Finally, we go back and calculate E to get 7.5. I understand the numerical difference is small, but the qualitative difference here is huge.

If people are interested, I can generalize the last step to give a general formula for any 2x2 non-recursive (so for example, GI's are recursive and are more complicated to calculate) game. And I can guarantee 100% you will see that any mixup with 1 safe option is always favorable, proven mathematically.

 

[MONEYMUFFINS]

 

[Nirf]

^ Note he said take the player out of the equation Nirf: this is pure mathematical theory, not how it actually works - its just an underlying guideline.

I understand that, but even as a mathematical theory it is fundamentally flawed. I don't mean simplified, I mean flawed with the 50/50 assumption. "Taking the player out of the equation" does not make it 50/50: it makes it so that we assume the two variables are uncorrelated. As I said before, Hates made two assumptions in how he calculated expected values, one was fully reasonable in the context of a simple theory, the other was not.

If you take players out of the equation alltogether, it is just a 50:50 and mathamatics can only treat it for what it numerically represents, so it has to assume 50:50 to literally mean 50 one way 50 another.

Outcomes, choices etc do not need to be equally probable. It's not "just a 50/50", it's just a situation with two possible decisions. To derive an expected value for this situation, we need to assume something about how often Asta will bullrush/grab, and similarly for Cervantes. 50/50 is an assumption just like any other, just a lot more poorly justified. Assuming that different outcomes have equal probabilities, while often the case, is also often not the case.

snip

Well, you actually did assume that each outcome resulted 50%, not just that each player picked 50% correctly. I do appreciate where you are coming from with try to simplify things. However, to quote someone famous "A theory should be as simple as possible, but no simpler". With the model you are suggesting, you are missing out on very fundamental ideas, in particular the notion of safety in half the mixup. That's my basic response to your points 2 and 3: yes of course I'm not taking everything into account with my model, but I'm taking enough into account that you see basic important points emerge. To be honest, what you do with the 50/50's is just adding and subtracting; the numbers you come up with are just sums of the characters damage outputs and don't tell you anything about the decision making matrix going on. Let me be clear: your post is very enlightening and well written. It's just that the math, at the level you've done it, in my humble opinion, adds absolutely nothing to the post.

Safety in half the mixup is a big deal: this idea is why sometimes in SC4 you will see BB go back and forth quite a few times between pairs of players: because it's reasonable damage but more importantly it's extremely safe. It's safer than TAS B in fact, because it spaces far better and gives better frames. And you do see lots of high level Soph's doing BB quite a lot, despite TAS B's massive damage advantage (yeah I know there are other reasons too before a crowd of people jump on me).

If you want Hates, I can derive a generalized "mix-up formula" which like I said will apply to a very broad swathe of mixups in the game (any 2x2, non recursive mixups) and people can apply this formula without worrying about the derivation. I'll do it tomorrow or something.