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Game Theory of Tech-Chasing in Super Smash Bros. Melee

In a little over two months, the party game known as Super Smash Bros Melee will become a 17 year old game (that’s old!), a game whose active community is still a prominent figure in competitive e-sports today. If you’ve ever seen Melee game-play, you may have realized that it is different from the average fighting game – no health bar, but a stage that serves as a giant “musical chair” off of which players try to force the other player. Besides that, Melee seems normal – if you move right on the joystick you run RIGHT, if you move up on the joystick you JUMP, and if you press a button, you may ATTACK. However, over the game’s long competitive life, the Melee community discovered difficult techniques/methods of controlling characters in ways that the video game designers did not intend; these imperfections, exploits, glitches, and other unintended options not only push the limit to how quickly characters can be controlled, but also allow a video game, which ultimately is just another computer program subject to limitations designed by the devs, to be the means of technical, creative, and perhaps even artistic expression.

Enough on the game, more on the game theory.

During a game of Melee, there is a particular moment known as a tech-chase opportunity, a situation to which casual players don’t really pay attention but with which competitive players are quite familiar. It occurs after the fighters have exchanged a few blows and taken some damage and begin to be knocked onto the ground (which doesn’t occur with little damage). When Player A hits Player B with a strong enough attack, it sends Player B hurling away, usually knocking them off the stage (again like musical chairs) or to the ground (if the attack isn’t strong enough). At the split second before Player B hits his/her back onto the ground (to trigger an animation of a dust cloud representing the impact), Player B’s range of motion becomes limited to a few options.

  1. No tech
  2. Tech Left
  3. Tech Right
  4. Tech in Place

Option 1 is the beginner option, because they fail to time it correctly or simply don’t know about it. If Player B does not tech, B is forced into a vulnerable, immovable state during which experienced players would take the opportunity to tack on more damage.

Player B can avoid this vulnerable “knocked-down” state by teching – simply tapping a button right before impact for the character to do a recovery roll on the ground. This technique is much quicker than waiting for the character to recover from its vulnerable state to get up. HOWEVER, teching is still exploitable because the opponent can predict which option might be chosen, bringing game theory into the picture.

Image result for tech chasing gif

 

In this gif, the Fox player mixes up his techs numerous times in order to try and get away from the opponent. However, the Falcon player correctly predicts and punishes his choice accordingly.

Teching is exploitable. Player B has a finite number of options to choose from in order to escape the oncoming assault. As a result, the assailant Player A must guess Player B’s choice to make to extend his/her combo. Here’s what I got for a payoff matrix:

Player B
Player A N L R I
N 1,0 0,1 0,1 ??
L ?? 1,0 0,1 0,1
R ?? 0,1 1,0 0,1
I ?? 0,1 0,1 1,0

N: no tech

L: tech left

R: tech right

I: tech in place

??: These cases are kind of difficult to explain because they differ depending on skill level, situation, etc. (Maybe in another follow up blog post …)

NOTE: Player B is trying to choose an option different from Player A’s. Player A is trying to guess the same one as Player B.

If we ignore the No tech rows and columns (resulting in a sub matrix as follows):

Player B
Player A L R I
L 1,0 0,1 0,1
R 0,1 1,0 0,1
I 0,1 0,1 1,0

This is similar to the Colonel Blotto game, but instead of two mountain passes to guard/attack, there are three. This makes it easier for Player B to escape and harder for Player A to succeed.

A pure strategy approach will not work because once Player A knows what Player B will do (suppose Player B will choose tech left), A can choose the option that will maximize his/her pay off (cover B’s tech left with an attack). Similarly, if Player B knows what Player A will do (suppose Player A will choose tech right), B can choose his escape options (tech left or in place to escape).

Instead, Players A and B want to make their choices unpredictable; that is, choosing a mixed-strategy approach enables them to randomize their options more. In reality, players often will have a propensity to choosing one option to another, but that’s where the creative mind-games begin like conditioning your opponent to choose one option over time to predict it when it’s most necessary. Conditioning is interesting because it is a method of inducing a bias for an option. In other words, over the entire game, biases may develop for one option allowing for one player to exploit it at the most necessary opportunity (I haven’t learned about dynamic models yet :\).

I want to end this discussion by mentioning the assumptions and short-comings to my cases.

This model assumes that each option has equal opportunity of escape/reward (depending on which player you are). In high-level, fast, dynamic game-play, parameters like stage position, character strengths, character weaknesses, and damage taken, take part in varying the payoffs in the matrix. For instance, if Player B is forced near the left ledge of the stage, teching left and teching in place are essentially the same. This is because teching left next to the left ledge won’t push the character off the ledge but just up to the ledge, which would essentially be the equivalent to teching in place. In this case, the two options, tech left and in place, merge into almost one option. Now if Player A recognizes this, he knows that he can cover two options at once rather than one. Reciprocally, Player B must change his probabilities in accordance with this, perhaps choosing 50% tech right and 50% tech left or in place. Character strengths and weaknesses are also crucial to the discussion. Depending on which character Player A is playing as, Player B may choose options that make it difficult for Player B to punish. If Player A chooses a slow character, maybe choosing to tech “away” rather than “in place” or not at all is optimal. If Player A chooses a character that has strong attacks forward but weak ones behind, then perhaps teching behind the opponent is better.

In Melee, there are many archetypes for the players: the naturals (understand the game at a fundamental level), the technical and flashy superstars (incorporate difficult techniques to optimize, confuse, or just disrespect), the mind-readers (condition or exploit opponent’s habits), and the robots (practice a few simple things and take it to the next level). Notably, a player named Wizzrobe falls a bit under the robot archetype as he has mastered tech-chasing. He has rehearsed the tech-chase situation to a level such that the opportunity is no longer about guessing, but rather about reaction time. Top players like Wizzrobe bypass the game theory entirely and can react to the choices, eliminating the probability aspect of the situation. Wizzrobe makes it a pure-strategy game in which he knows which choice will maximize payoff, i.e. the one that his opponent chooses.

There are also trees of other options that come from choosing to not tech like “get-up attack,” which I haven’t addressed in this post. But Melee is a dynamic game of game theory, with tech-chasing just being one of the countless ways game theory is woven into the video game. Players who have more experience typically are better because they’ve learned to recognize more situations as a game theory moment. Analyzing static situations – like this tech-chase opportunity – can inform what a beginner player should do in such dilemmas and ultimately make him/her a better player.

I like what he says around 4:45.

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