Game Theory

Warning: theory fighter ahead, don't bother reading or replying if you're just gonna say 'dude, that doesnt matter to the real game.'

So I read 'The Compleat Strategyst' the other day, and had been thinking about how game theory applies to VF, which brought up a question: How much do you value frame advantage or disadvantage?

At a simple level, the value of a successful attack or throw is obvious - it's the amount of damage done. e.g. a basic model of a nitaku / 2-choice situation where you choose between Throw (say, 40 pts) and a mid Attack combo (say, 60 pts); while your opponent chooses between Guard or a reverse nitaku Attack would look something like this:
_| A G
A 60 0
T -60 40

Meaning if you choose Attack while they choose Attack, you get 60 pts of damage, Throw vs Attack you lose 60 pts, and so on.
So you should choose A about | -60 - 40 | = 100 times for every | 60 - 0 | = 60 times that you choose T, or a 5:3 ratio between attack and throw. On average this nitaku will net you about (5 * 60 + 3 * -60) / 8 = 15 points of damage; so being on the advantage end of this nitaku is 'worth' 15 points to you.

Of course, this model isnt realistic - for instance, when you choose A and the opponent chooses G, the situation isn't really 0 pts of damage; you're at a disadvantage even if no damage is guaranteed. How much is that disadvantage worth?

This is easy to figure if the frame advantage you have going into the nitaku is identical to the frame advantage the opponent has if she blocks. E.G. jacky vs. jacky, you block and respond with either throw or ; you're at +7fr going in and -7fr if you choose A vs G. So what's the Value of 7 frames?

_| A G
A 60 ?
T -60 40

With the same characters, being at -7fr is worth the same as being at +7fr, just with an opposite sign. But +7fr is the exact situation we're considering. So the value of the space marked ? (call it -V) is the same as the overall value of the nitaku itself, and you have

_| A G
A 60 -V
T -60 40

V = ( 100 * 60 + (60 - -V) * -60 ) / ( 160 + V )

which works out to about 10.5. Thus, once you consider frame disadvantage, this nitaku is only worth about 10-11 points instead of 15; you should throw more often than previously calculated, about a 10:7 ratio between attack and throw.

The worth of 7 frames was easy to calculate because of the identical choice of character and attack. It should be similarly easy to calculate the value of some guaranteed situations, e.g. if you're evo jacky and at +13 frames, i'd be willing to bet , dominates all other options, there's not much value in throwing or risky attacks.

My question is, for other non-trivial frame disadvantage / advantage situations, what rule of thumb would you use to decide how much those frames are worth?

 

Alot goes on in any vf match that has little to do with advantage or disadvantage. I think to exclude "magics" from any calculations you make (and from the characters you use!) is to ignore a big part of the game.

Then, of course, there are options available the defender in nitaku situations apart from guard & attack as well as options available the attacker apart from mid & throw. To ignore this "other stuff" is to reduce vf to something unrecognizable as vf.

In order to avoid the pitfall of being too reductionist, look more closely at specific moves and matchups like your jacky situation. What is it worth to you to block akira's open stance DE using lei fei? There's so many things to think about in a situation like this...akira can attack, block, ETEG, ARE, E>attack, Inashi & the list goes on with different risks & rewards. While I question the value of going through all sorts of calculations to arrive at an abstract numerical equivalent for something I already have a concrete numerical equivalent for & which I can feel while playing, I would be interested to hear exactly how many points of damage my +6 (or whatever) is worth.

have fun!

 

Yeah, the actual decision matrices are bigger than 2x2, but they're still a manageable size; most players are choosing from maybe 5-6 options for most situations. Furthermore, I bet that certain strategies dominate. E.g. at -8, assuming you have the skill, TEG dominates G. If I stay interested in this i'll probably whip up a script to solve larger matrices for more interesting situations.

I assume by 'magics' you mean odd consequences of animation and hit detection on certain moves that allow you to beat faster moves from disadvantage? Even those seem to be based on pretty well defined factors such as stance, so you can just do a matrix for each stance and et voila, jacky should do ~ more often in open stance than closed. Its not tekken or soul calibur; there may be some fuzzy edge cases but its a mostly rational game. Of course, if you mean the mental aspect (noticing an opponent's suboptimal strategy and punishing) I totally agree that its possible to be too reductionist; if it was math fighter no one would play it /versus/images/graemlins/wink.gif

Reasoning about range and position effectively would be harder, for sure. (near the edge? TFT's value now = 200 ? ) Still, I think the outcomes of close range decisions (away from walls, etc) do boil down to either damage or advantage/disad. What else do you gain from an exchange?

The real question is how to place a value on the outcomes - You say you already have a concrete numerical equivalent for your +6; what is it and how did you come up with it? I mean, sure, its somewhere between 0 dmg and 32dmg (assuming first time you get guaranteed dmg is at ,), but where?

 

[ QUOTE ]
s_aki said:
What is it worth to you to block akira's open stance DE using lei fei? There's so many things to think about in a situation like this...akira can attack, block, ETEG, ARE, E>attack, Inashi & the list goes on with different risks & rewards.

[/ QUOTE ]

Apparently, Lei at +3 against akira after blocking DE is worth about 2.5 points of damage /versus/images/graemlins/wink.gif Seriously, you came up with the example, so I tried it . . . and came to suspect you manufactured it to point out some 'magics'.

First, the question of assigning values to outcomes.
Damage:
I used actual training mode damage for combos / counters where appropriate (bit of a pain in the ass, really; anyone know exactly how scaling works?).
I initially assumed average value of Leifeis throw damage for a successful throw (50pts) and 1/2 of that for a throw attempt against 2xTE (25pts).
As a check on that, I did a matrix just for the sub-game where lei picks one of 4 throws and akira does 2 te; if you assume a value of 0 for a successful escape (-6 frames, more on this later) it turns out that the average value of that game to Lei is indeed 25 points.

Advantage/disadvantage:
So if mid-large adv (non-ThG) is worth 0, a guaranteed throw attempt against 2xTE is worth about 25 points.
I reran the throw matrix assuming non-ThG is worth about 10 (say, P / low P); then, a guaranteed throw against 2xTE is worth about 20 points.
Thankfully, this was all the analysis of advantage needed in the lei vs akira case. Even more thankfully, it turns out that the difference between those two views of adv / disadv affects the optimal strategy but does not affect the value of the game.

Which attack / defense options to consider? I'm not a akira or leifei player, but this is what I used:

Akira: ( I left out ARE from your list because he has to make that decision before the DE gets blocked)
A. Attack (dblpalm, altho yoho might be interesting)
B. Guard
C. E-2TEG
D. E-dodge attack
E. ++, ( I assume thats what you meant by inashi)

Lei:
W: Throw
X:
Y: +
Z:
Other options that came to mind (eg lowpunch) were dominated by

Decision matrices:
First, the thG = 25, anything less = 0 approach:
(akira's choices on top, leis on left; positive is good for lei)
__| A | B | C | D | E
W -37 50 25 -25 50
X 56 00 21 -25 -70
Y -36 -25 -25 54 44
Z 30 00 00 -25 20

There's definitely some 'magics' here; X is leis' full circular but akira's dodge attack to his front foot will beat it and his option Z , but Y + tracks and smacks it.
The -70 for inashi f+P, P, P, dlc is a guess b/c my scrubby ass could only get the first punch in before the dlc, for 69 points. /versus/images/graemlins/wink.gif

So anyway, the value of this matrix is that lei comes out ahead with 2.5 points of damage against akira. This sorta makes sense to me, cause akira is still a beast at -3 frames.

An Optimal strategy for lei:
Throw: 22%
: 36%
+:34%
: 8%

for akira:
dblpalm: 28%
guard: 33%
E-2TEG: 0% (!)
E-dodgeattack: 31%
inashi~ DLC: 8%
I'd be curious to see if this (!) makes sense to akira players . . . I probably undervalued a dodge compared to block, b/c without recovery time info its hard to know how much a successful dodge is worth.

Now if med-lg is worth 10 pts and thG worth 20

__| A | B | C | D | E
W -37 50 20 -25 50
X 56 00 21 -25 -70
Y -36 -20 -20 54 44
Z 30 -10 -10 -25 20

Here, the value of the game is still 2.5 points, but the strategies are slightly different:

for lei
Throw: 21%
: 36%
+:34%
: 9%

for akira:
dblpalm: 32%
guard: 29%
E-2TEG: 0% (!)
E-dodgeattack: 26%
inashi~ DLC: 12%

So it seems like as long as assumptions about the value of frame adv /disadv are consistent and within bounds, it shouldnt affect valuation all that much, even if it does affect strategy. And If nothing else, I learned more about akira and lei . . .

 

Replace Lei 3k with 66p, Lei 3k sucks. And what about doing nothing/boxstep at advantage? I hate to shit in your cereal but making all these charts and stuff may skew your tactics in favor of theory fighter stuff that really doesn't translate well to real matches.

 

Actually with dodge>attack I was thinking body check cause I'm goofy like that...hence, that akira's dodge attack beats throws at sm/med advantages & full circulars in some situations didn't factor into my thinking. Things change a bit with your matrices substituting evade>dbc...

The choice of de closed stance with lei was just sort of random. My approach with akira lately has been (as much as possible) keep stance in mind & watch constantly. Constant attention to stance & using de & sde (gotta love evo) makes it easier to get full damage for those combos and move my opponent around the ring. It makes it easier for me to exploit chronic dodging to one side or to yomi dodges aiming to avoid spod or dfs. Basically it gives me as clear a set of choices with each exchange as I can have. The more clear my choice set and control of offense I have, the more opportunity I have to find the big damage, yomi based options like that inashi'd high kick (or spoding/ing a bad throw)...it also helps me make shrm magic.

When I talked about a concrete numerical equivalent, I meant +6...the six frames I often throw away to yoho a dodge. This +3=2.5 reminds me of how they have computers evaluate chess positions in terms of +/- fractions of a pawn.

And sure, with no delay to exploit the failed dodge, eteg is going to be your reset option and should get you 00.

I think the 10/20 approach is closer to the way people regard things when they play.

Dandy...I don't think the point is actual matches for KoD as a player. As I understand it, the point is evaluating the choices two players have when playing the game and what value to them non throw or attack guaranteed situations are to them damagewise. My thinking is that +6 will mean more for Jeffery v the cast than it will for, say, aoi v the cast and coming up with the numbers that eventually support notions like this would be an involved process to say the least.

Maybe going throw some actual matches (evo disk, slow mo) and working up the exchanges for some practical data would be interesting to you, KoD?

 

Dandy: thanks for the suggestion. It doesn't change lei's strategy much, but does raise the value of the game to about 2.7. Makes me wonder if 1 frame = 1 point would be a workable rule of thumb from 1-7 frames. Strictly speaking, 'doing nothing' at advantage (i.e. neutral joystick, no button inputs) is a bad idea unless you have an auto-inashi; what exactly did you have in mind, and how would it fare against akira's options? As for shitting in my cereal, what part of 'warning, theory fighter ahead' did you not understand? /versus/images/graemlins/wink.gif

s_aki: yeah, i'd say your description of my point is pretty accurate - I dont expect this to suddenly make me a better player or anything, but it's interesting to me. Its especially interesting from a game design point of view - both sides have some high damage options, but sega balanced it to the point where on average akira in disadvantage really is going to take a slight bit of damage. Your comparison to valuation of chess pieces / positions is also right on. That stuff is occasionally useful to me playing chess (should I trade a knight or a bishop?) but it's not the whole game.

I will say however that people's experience and intuitions about games, probability, etc. are quite unreliable. E.g. the birthday paradox - you have 10 people in a room, whats the chance one of them shares a birthday? 10 people, 365 days, should be around 1 in 36 right? It wouldn't suprise me if common intuitions about VF were also a bit off . . .

. . . like, for instance, ETEG being the safe, reset to 0 option. I would have figured the same thing. But the reason I put an exclamation mark there is because, with these options, it doesnt net akira 0dmg, it nets him over -5 dmg against lei's optimal strategy. In other words, its such a bad option that its irrational to ever do it.

I probably undervalued a successful ETEG, but in order for it to be as worthwhile as just guarding, we'd have to say that a successful ETEG against lei's or is worth more than 21 points of damage, which seems a bit high to me (esp. since akira has to be holding down guard at the end of his 17~19 frame evade). Again, without recovery info I don't know though - does the orange book or any of the other mooks have recovery frames?
Edit: just realized 1/60th is still available via the wayback machine . . . boy was I wrong about the recovery. Seems like is pretty well guaranteed for akira (or at least cpu wont block it), but I cant get much else that requires directional input b/c of the lack of buffer. So what's a good delay starter for lei?

As for the replays on the dvd, I had been thinking about that, but more from a point of view of 'if the supposed optimal strategy at -3 is X, does that match what Homestay actually does?'

 

I was thinking craft the analysis to what is practiced rather than analyse "in a bottle" as it were and see if it matches. I think homestay just wins with coldblooded invention, exectution and yomi.

I was thinking 1frame=1pt myself actually /versus/images/graemlins/wink.gif

 

[ QUOTE ]
KoD said:
Dandy: thanks for the suggestion. It doesn't change lei's strategy much, but does raise the value of the game to about 2.7. Makes me wonder if 1 frame = 1 point would be a workable rule of thumb from 1-7 frames. Strictly speaking, 'doing nothing' at advantage (i.e. neutral joystick, no button inputs) is a bad idea unless you have an auto-inashi; what exactly did you have in mind, and how would it fare against akira's options?

[/ QUOTE ]
66p also pushes back on block, negating a forced guessing game if it is guarded. As for doing 'nothing', take this for example. You have +4 or something, and you backdash -> evade. If the opponent attacks back, you can combo for free. If they guard or evade, you are in a neutral situation. Only a complete madman would dash in and throw at -4.

 

Yeah, boxstep raises the theoretical value for Lei to well over 3pts. It also makes ETEG even less attractive of an option for akira . . . i mean, i'd expect evading to be more useful at higher disadvantages, but its still odd.

Anyway, hopefully I'll have time over the weekend to whip up something that lets me automatically plug into frame data into a matrix for 'what if' scenarios.