I know it works best in zero sum games. For a contrived example say we are on offense and our opponents only options are to block high or low. Our only options are to attack mid or low. If our mid attack lands we get a juggle that leads to 110 damage points but if our mid is blocked they get 40 damage points. If our low hits we get 30 damage points. If it's blocked they get 80 damage points. To sum it up.


Mid hits=110 damage points mid is blocked we lose 40 damage points
Low hits =30 damage points low is blocked=we lose 80 damage points. Can we use math to figure the optimal ratio of mid attacks to low attacks?

we have to know the probability of the various attacks landing which depends on opponent and varies based on multiple factors including your previous ratio of the different moves

so no, you can't really math out that kind of fighting game decision

if you really want to apply math to fighting games, there are rough risk/reward calculations you can do

So we can't find a Nash for high low mixup

OK in the above situation our advantage is 35 points if we do an exact 50/50 and our opponent blocks at an exact 50/50 ratio

I think you could, in theory. But in practice, it's not going to help you.

There's a very decent article out there about applying GTO to super smash brothers melee. Sry no link tho.

If you were to break it down, you could say each frame in the game is a simultaneous move by both players and analyze that using game theory. But since there are (typically) 60 frames per second, the decision tree you are analyzing would be enormous. So it's not of much practical use even though mathematically it's possible.

In the specific scenario given, assuming that the payoff function is the net damage done, I'm getting the Nash equilibrium as you attack mid 42.31% of the time (11/26) and the opponent blocks mid 73.08% of the time (19/26). Not 100% sure if I did the algebra right but it's definitely possible to calculate with a bit of math to see that the payoff for those percentages is independent of the opponent's action (average payoff is equal no matter what the opponent does).

http://alexspuffstuff.blogspot.com.a...tive-play.html

Just curious what formula did u use? I only know how to solve simple things using a matrix.

I just wrote out the payoffs for each player and then set the payoffs equal to each other and solved using a bit of algebra.

If x = percentage of attacking mid and y = opponent percentage of blocking mid.

The attacking player's payoff function is
x*(110(1-y) - 40y) + (1-x)(30y - 80(1-y))

So the defending player's unexploitable strategy is the one where it doesn't matter what value the attacking player chooses for x, i.e. it's the one where
110(1-y) - 40y = 30y - 80(1-y)
110 - 150y = 110y - 80
190 = 260y
19/26 = y

so that's how I got 19/26 or 73.08% blocking mid as the unexploitable strategy. Same idea to find the unexploitable value of x as well, just using the opponent's payoff function. I think this method of setting the opponent's payoffs equal to each other works for all zero sum games.

TY

One thing that would immediately complicate this is even if the opponent is forced to eat a mixup they have multiple options Wether they block high or low. For example they may sacrifice damage on a blocked low for better oki opotions or wall carry