Melee Metagame as Payoff Matrices
I’d like to explore the moment a character grabs the ledge in Melee, but rather than deconstructing it into just a payoff matrix and trying to find a Nash equilibrium in it, I discuss about how knowledge of a payoff matrix can reveal a lot about the current meta of a current game.
Melee can be seen as a game of musical chairs in which the stage is the chair and players force the opposing character off of the stage. First person to do that four times in a competitive match is the winner, so players frequently find their characters holding onto the ledge after recovering from some powerful attack from the opponent that had sent them flying off stage.
(source: smashwiki)
The ledge is an interesting area on the stage as only one person may be grabbing the stage at a time. Also, grabbing the ledge will grant a brief moment of intangibility (~half a second), give a character their second jump back (every character can jump mid-air but only once… physics), and allow you to let go of the ledge with your double jump by tapping down. Half the battle of recovering from a powerful attack is trying to get onto the ledge because the opposing player will attempt to prevent you from doing so by edge guarding (preventing a character from getting near the stage to take a life). The other half is figuring out how to get back both feet firmly planted back on the stage.
Let’s set this situation up. Assume a sudden death situation in which A is on the ledge and B is ready to respond to A’s ledge option. Assume correctly choosing the right option for player B means winning the match and incorrectly choosing the right option means neither of them have won yet.
G: normal get-up
R: roll onto the stage
A: get-up attack
J: jump from the ledge
| Player B | |||||
| Player A | G | R | A | J | |
| G | 1,-1 | 0,0 | 0,0 | 0,0 | |
| R | 0,0 | 1,-1 | 0,0 | 0,0 | |
| A | 0,0 | 0,0 | 1,-1 | 0,0 | |
| J | 0,0 | 0,0 | 0,0 | 1,-1 | |
This is quite a boring payoff matrix and doesn’t adequately describe the complexity of being stuck at the ledge. This also assumes that the choice of Player B is 100% commitment. It is quite hard to find such instances of 100% commitment because Melee is played more as an option coverage game rather than choice. One thing to take into consideration is whether choosing one option can also cover or set-up for another option but with not as optimal of a punish. Assuming that these suboptimal reaction based punishes have a certain probability of succeeding…
| Player B | |||||
| Player A | G | R | A | J | |
| G | 1,-1 | (1-p), p | (1-p’), p’ | 0,0 | |
| R | (1-p”’), p”’ | 1,-1 | (1-p””), p”” | 0,0 | |
| A | 0,0 | (1-p””’), p””’ | 1,-1 | (1-p”””), p””’ | |
| J | (1-p”””’), p”””’ | 0,0 | 1-p””””), p”””” | 1,-1 | |
This also is not a fun matrix: too many probabilities and variables. This is also not a correct matrix. But the takeaway is that these variables are the ones that makes Melee interesting. Characters like Fox and Sheik are great for guarding the ledge with because their quick mobility allows them to perhaps cover more options at once or take full advantage of a player’s reactions. They are also great to get off the ledge as their frame data are exceptionally good, enabling them to defend quicker. Stronger characters like Ganondorf may have higher probabilities of winning because even his sub-optimal attacks may be enough to kill the opponent. It turns out the characters that are both fast and strong – that is, those that have better payoff matrices for ledge guarding – are better at ledge guarding.
But as new techniques were discovered – wave dashing, wave-shining, pillar combos, Ken combos, wobbling, knee… etc – characters that used to be considered top tier early in the game are low tier characters today. This allowed characters to have access to tools that they couldn’t before. Perhaps it is because the discovery drastically alter the payoff matrices of a character. For instance the discovery of ledge-dashing, an advanced technique that, when performed optimally, can grant intangibility for a character, adds an additional option to the above payoff matrix. This technique benefits some characters more than others, especially Fox, who gets the most amount of intangibility from executing it well. This discovery must have changed the payoff matrix, changing the meta such that Fox is even better before the discovery.
Despite having been released 17 years ago, debates about which character is better, which stage is more beneficial for a certain characters, and matchups continue. Is Fox or Falco better? Is FD the best stage for Marth? Is Marth vs Fox 60 40? Suppose data for the correct payoff matrices of every kind of micro-interaction for every character at every stage against every character (i.e. recovery at Yoshi’s with Falco against Marth, tech-chasing with Mario against a Jigglypuff, etc.) was posted on a website today. There would be a quantifiable measure of how good one character is to another. For instance, given the payoff matrices for Fox and Falcon for neutral exchanges, there would be quantifiable measures of which character has the upperhand. The biggest flaw in this hypothetical is figuring out what method should be used to quantify the probabilities. Would it be based off of what is capable (TAS level)? Or would it be based off of the history of all the matches played so far in the game. How do you quantify something as variable as a player’s reaction time? Nevertheless, given such database of measures, a metagame is defined no longer with just pro player opinions but with numbers, something that may be quite interesting to have in the future.