Mixed Equilibrium and Insane Sequences in Competitive Pokémon Singles
The Pokémon game franchise has been a favorite for audiences of all ages over its incredibly long 24-year old lifespan. Undoubtedly, this sustained popularity is in part due to the easy-to-learn, but hard-to-master battle system. In very basic terms, one’s goal in each battle is to faint/knock out all six opposing Pokémon by reducing their hit points (HP) to zero. In singles, each Pokémon is sent out one by one. The combat is turn-based: both players select a move (without knowing what the opponent chooses) at the start of the turn, and then both moves are executed. Some Pokémon have an inherent advantage over other specific Pokémon because their type is advantageous against that specific Pokémon, or simply because they are faster and can deliver a finishing blow before their opponent can respond. Naturally, to circumvent this, the player at a disadvantage can switch out their Pokémon into a matchup that is more favorable for them. However, the opponent can react with a setup move to boost their attack, taking advantage of the weaker player using their action to switch.
The simplicity of it means that predicting your opponent’s moves lies at the heart of competitive singles Pokémon. To explain this further, Sagona presents an example of this common dilemma in a battle:
Let’s look at the following example from Pokémon:
You have a weakened Charizard against a Lucario. If
the Charizard stays in it is killed by Lucario’s Quick Attack.
But if Charizard switches out to avoid the Quick
Attack, you risk Lucario getting off a Swords Dance. So
do you predict Lucario’s Swords Dance by attacking him
with Flamethrower – or not risk losing your Charizard to
a Quick Attack? (Sagona, 1)
In this case, there is a game that clearly does not have any dominant strategies. If I, as the player controlling Charizard, were to always switch out, then my opponent would always respond by using the setup move Swords Dance, gaining a massive advantage. Then, I would naturally be more inclined to attack using Flamethrower. But if my opponent catches wind of that, he will simply attack with the faster Quick Attack and knock me out. So, this situation clearly demonstrates a mixed equilibrium, similar to the penalty kick scenario we discussed in class.
Although, knowing this naturally begs the question as to how to represent the payoffs in a game of Pokémon. Sagona proposes a framework that I think encapsulates what each player stands to gain from their decisions:
Payoff Table for Charizard vs. Lucario, from the perspective of Charizard (Sagona, 2)
In this case, Sagona uses the net amount of Pokémon advantage each player would have after resolving the move, from the perspective of the Charizard user. A +1 represents knocking out an opposing Pokémon, while a -1 indicates losing a friendly Pokémon. The -3 from Swords Dance comes from the fact that a Lucario with boosted attack can easily knock out three friendly Pokémon (Sagona, 2). Though the values of payoffs simplify things (e.g. the Pokémon switching in would still take damage, causing a fractional net loss), but it successfully captures the effects of the decision in our scenario. This starting point would allow us to calculate the mixed Nash Equilibrium by supposing that the expected payoff from each option must be equal. Sagona does this, and calculates that Charizard should use Flamethrower 3/5 of the time, and Lucario should use Swords Dance 1/5 of the time (2-3).
Despite giving the highest expected outcome by definition, real players are obviously not driven entirely by this Nash Equilibrium. In fact, the game is most exciting when players deviate furthest from the probabilities that the Nash Equilibrium dictates. For instance, in the 19th game of the World Cup of Pokémon series between India and USA South, Freezai pulls off a borderline insane play. Starting on Turn 30, Freezai is faced with the most critical situation of the game. Crawdaunt, the primary “wallbreaker” of Freezai’s team, is up against Gastrodon, the Pokémon holding Eo’s defense together. Freezai has two viable options. He could use Crabhammer, which not only does zero damage but boosts the Gastrodon’s special attack to allow for his opponent to knock him out. The other option is Knock Off, which would knock out the Gastrodon, a critical defensive member of Eo’s team. If Eo switches out his Gastrodon, Crabhammer would knock out any of the other members, while Knock Off would do comparatively very little. From this, we can construct a payoff grid similar to Sagona’s:
Payoff Table of Crawdaunt vs. Gastrodon, from POV of Crawdaunt, constructed in style of Sagona
Note that instead of +1, the gain from (Crabhammer, Switch) is only 0.9. This is because Crabhammer actually only has an 90% accuracy! Even against the odds, he manages to use Crabhammer three times in a row, in the exact same critical situation vs Gastrodon, which ends up securing the victory.
In conclusion, we learned that mixed Nash Equilibrium indeed maximizes the expected value of the outcome. However, there are often many factors that contribute to each player’s decision in a real, competitive setting. The example that I provided illustrates that despite the existence of a Nash Equilibrium, there are approaches to the game that have incredibly high risks, but also incredibly high rewards. This unpredictability keeps us on the edge of our seats, even if the actual game is simple.
Sources:
Sagona, Steven. The Game Theory of Pokemon: AI implementing Nash Equilibrium. 12 May 2014. https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&cad=rja&uact=8&ved=2ahUKEwjB5ZOPx6L6AhUcD1kFHZPFBggQFnoECAoQAQ&url=https%3A%2F%2Fsmashboards.com%2Fattachments%2Fgame-theory-latex-pdf.90707%2F&usg=AOvVaw3f97HHR5SqK4PfUKntSaTa.
Freezai vs. Eo