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The Game Theory of Melee

blue cheez

Smash Cadet
Joined
Jan 12, 2007
Messages
44
Hi everyone,

There's been quite a bit of discussion in the Peach forums about how to deal with "rock,paper, scissors"-like situations that peach often encounters. I've mentioned on the forums before how these situations can be thought of as "simultaneous games" which can be solved using game theory, but I thought it might be helpful to the community to have a write-up that introduces game theory and goes over how it can be applied to melee.

The writeup is attached as a PDF.

I originally wrote a two-page short summary, but discovered that everyone who read it didn't really understand it.
Instead I decided to do a full comprehensive writeup of the topic - one that hopefully won't leave anyone with the patience to read it confused.

To the impatient melee player who doesn't want to read 16 pages:
I encourage anyone who is mildly interested to read the first four and a half pages. If you just read those four and a half pages, I think you will have a sufficient understanding to be able to apply some game theory on your own!

It's been quite a few months since I said I would do this write-up. So I'm rushing it out even though I'm not quite done with this whole gametheory analysis of peach thing. The write-up itself is complete. It introduces and explains game theory as it applies to Melee. But I only briefly mention the more complicated frame advantage situations that I've been analyzing.

A full analysis of these special, peach-specific situations will come soon, as a second write-up.
 

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Quetzalcoatl

Smash Ace
Joined
Jun 4, 2003
Messages
622
Location
Perth, Western Australia
There is a specific situation I'd like a game theory analysis of:

Missed Rest Punish

When Puff misses a rest, Peach players are currently relying on up throw turnip combos to punish, however as we know there is always a way for the Puff player to escape this and its never true.
What I want to know is if the payout of going for this unreliable punish is better or worse than 2 other options:
- Charged Fsmash (club)
- Searching for a stitchface

To clarify here are the 3 competing strategies in detail

1. Turnip combo punish that can lead to about 35% damage, unreliable (players can DI out of the combo)
2. Fully charged golf club (24%). Club is preferred over Pan because you can reliably hit with Club by spacing and filtering
3. Searching for a special turnip (dot eyes and above), can comfortably pull 4 turnips while Puff is sleeping, can pull 6 if knitting is used. FC Aerial is used in the event of an unsuccessful search (14%)
 

blue cheez

Smash Cadet
Joined
Jan 12, 2007
Messages
44
So the game theory that I'm doing the writeup is for "simultaneous games," which are situations where you and your opponent cannot simply react to eachothers options but are forced to guess. Basically weird rock-paper-scissors games. You don't need any of that game theory here to decide which option is the best option. The reason why this is is because the opponent can easily see which option the opponent has chosen and counter it. So there's no rock-paper-scissors going on between players. Finding the optimal option is as simple as calculating the average amount of damage you'll deal with each possible outcome:

1. Turnip combo punish that can lead to about 35% damage, unreliable (players can DI out of the combo)
1. I'm not sure what "turnip combo punish" you're referring to. Do you mean throwing a turnip in the air and combo'ing them with downsmash? Can you also tell me what the opponent is doing to get out of the combo?
2. Fully charged golf club (24%). Club is preferred over Pan because you can reliably hit with Club by spacing and filtering
2. What stops you from hitting them into a falling turnip for an extra 6%?
3. Searching for a special turnip (dot eyes and above), can comfortably pull 4 turnips while Puff is sleeping, can pull 6 if knitting is used. FC Aerial is used in the event of an unsuccessful search (14%)
3. This is very easy to calculate by hand. You just find the average amount of damage this will do.
average damage = (14%-damage)*probability not getting (dot/stitch/etc) + (35%-damage) times probability of getting stitch
+(16%-damage)*probability of getting dot + 35% times probability of getting a bomb.

If you took a college level probability course, you could find these probabilities using the binomial distribution:
Probability of getting "s" stitches in "t" tries = (t choose s)(Probability of stitch)^s(Probability not stitch)^t
= t!/(s!(t-s)!)(1/58)^s(57/58)^(t-s) = 4(1/58)(57/58)^3 = .065

If choose functions intimidate you, you can do a highschool probability trick to find these probabilities:
Probability of getting a stitch in 1 pull = 1/58
Probability of not getting a stitch in 1 pull = 1 - 1/58 = 57/58
Probability of not getting a stitch in 4 pulls = (1- 1/58)^4 = (57/58)^4
Probability of getting at least 1 stitch in 4 pulls = 1 - probability(not getting a stitch in 4 pulls) = 1 - (57/58)^4 = .067
Don't be worried that this is a slightly higher number than before. Before we found the probability of exactly 1 stitch, while here we foudn the probability of at least 1 stitch. So we're including the probabilities of grabbing 2, 3, and 4 stitches as well (which are small enough that this is an OK approximate of what we did above).

You can do the same thing for dot and bomb, as well. But for now I'm only going to consider stitch as our only important item (the others don't affect the numbers very significantly, but you can check it yourself if you dont believe me)
so now we have:
average damage = (14%)(prob not stitch in 4 pulls) + 35% (prob stitch in 4 pulls) = 14(.935)+35(.065) = 15.4
for knitting average damage, you get 2 extra tries so: prob stitch = 6(1/58)(57/58)^5 = .095
average knitting damage = (14%)(prob not stitch in 4 pulls) + 35% (prob stitch in 4 pulls) = 14(.905)+35(.095) = 16.0
Knitting doesn't help you by much more; you only squeeze about half of a percent by doing it.


My suggestion is that the optimal punish is likely a mix-up of two different turnip combos that come out so fast that the opponent has to guess which option peach will choose (turn the punish into a rock paper scissors game).
So imagine peach throws up 2 turnips, and either downsmashes OR fsmashes (you can't charge it though because then your opponent can SEE you charging it and counter that specific option). Now the opponent is forced to guess between two different DI options, and you might be able to squeeze out more percent out of the situation on average. Only then can you do the "simultaneous game theory" stuff in the writeup.
 
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