Mathematical research of Pokémon

Solution 1:

Has there been serious research? Probably not. Have there been modeling efforts? Almost certainly, and they probably range anywhere from completely ham-handed to somewhat sophisticated.

At its core, the game is finite; there are two players and a large but finite set of strategies. As such, existence of a mixed-strategy equilibrium is guaranteed. This is the result of Nash, and is actually an interesting application of the Brouwer Fixed-Point theorem.

That said, the challenge isn't really in the math; if you could set up the game, it's pretty likely that you could solve it using some linear programming approach. The challenge is in modeling the payoff, capturing the dynamics and, to some degree (probably small), handling the few sources of intrinsic uncertainty (i.e. uncertainty generated by randomness, such as hit chance).

Really, though, this is immaterial since the size of the action space is so large as to be basically untenable. LP solutions suffer a curse of dimensionality -- the more actions in your discrete action space, the more things the algorithm has to look at, and hence the longer it takes to solve.

Because of this, most tools that people used are inherently Monte Carlo-based -- simulations are run over and over, with new random seeds, and the likelihood of winning is measured statistically.

These Monte Carlo methods have their down-sides, too. Certain player actions, such as switching your Blastoise for your Pikachu, are deterministic decisions. But we've already seen that the action space is too large to prescribe determinism in many cases. Handling this in practice becomes difficult. You could treat this as a random action with some probability (even though in the real world it is not random at all), and increase your number of Monte Carlo runs, or you could apply some heuristic, such as "swap to Blastoise if the enemy type is fire and my current pokemon is under half-health." However, writing these heuristics relies on an assumption that your breakpoints are nearly-optimal, and it's rarely actually clear that such is the case.

As a result, games like Pokemon are interesting because optimal solutions are difficult to find. If there were 10 pokemon and 20 abilities, it would not be so fun. The mathematical complexity, if I were to wager, is probably greater than chess, owing simply to the size of the action space and the richer dynamics of the measurable outcomes. This is one of the reasons the game and the community continue to be active: people find new ideas and new concepts to explore.

Also, the company making the game keeps making new versions. That helps.

A final note: one of the challenges in the mathematical modeling of the game dynamics is that the combat rules are very easy to implement programmatically, but somewhat more difficult to cleanly describe mathematically. For example, one attack might do 10 damage out front, and then 5 damage per round for 4 rounds. Other attacks might have cooldowns, and so forth. This is easy to implement in code, but more difficult to write down a happy little equation for. As such, it's a bit more challenging to do things like try to identify gradients etc. analytically, although it could be done programmatically as well. It would be an interesting application for automatic differentiation, as well.

Solution 2:

I think it's worth pointing out that even stripping away most of the complexity of the game still leaves a pretty hard problem.

The very simplest game that can be said to bear any resemblance to Pokemon is rock-paper-scissors (RPS). (Imagine, for example, that there are only three Pokemon - let's arbitrarily call them Squirtle, Charmander, and Bulbasaur - and that Squirtle always beats Charmander, Charmander always beats Bulbasaur, and Bulbasaur always beats Squirtle.)

Already it's unclear what "best strategy" means here. There is a unique Nash equilibrium given by randomly playing Squirtle, Charmander, or Bulbasaur with probability exactly $\frac{1}{3}$ each, but in general just because there's a Nash equilibrium, even a unique Nash equilibrium, doesn't mean that it's the strategy people will actually gravitate to in practice.

There is in fact a professional RPS tournament scene, and in those tournaments nobody is playing the Nash equilibrium because nobody can actually generate random choices with probability $\frac{1}{3}$; instead, everyone is playing some non-Nash equilibrium strategy, and if you want to play to win (not just win $\frac{1}{3}$ of the time, which is the best you can hope for playing the Nash equilibrium) you'll instead play strategies that fare well against typical strategies you'll encounter. Two examples:

• Novice male players tend to open with rock, and to fall back on it when they're angry or losing, so against such players you should play paper.
• Novice RPS players tend to avoid repeating their plays too often, so if a novice player's played rock twice you should expect that they're likely to play scissors or paper next.

There is even an RPS programming competition scene where people design algorithms to play repeated RPS games against other algorithms, and nobody's playing the Nash equilibrium in these games unless they absolutely have to. Instead the idea is to try to predict what the opposing algorithm is going to do next while trying to prevent your opponent from predicting what you'll do next. Iocaine Powder is a good example of the kind of things these algorithms get up to.

So, even the very simple-sounding question of figuring out the "best strategy" in RPS is in some sense open, and people can dedicate a lot of time both to figuring out good strategies to play against other people and good algorithms to play against other algorithms. I think it's safe to say that Pokemon is strictly more complicated than RPS, enough so even this level of analysis is probably impossible.

Edit: It's also worth pointing out that another way that Pokemon differs from a game like chess is that it is imperfect information: you don't know everything about your opponent's Pokemon (movesets, EV distribution, hold items, etc.) even if you happen to know what they are. That means both players should be trying to predict these hidden variables about each other's Pokemon while trying to trick the other player into making incorrect predictions. My understanding is that a common strategy for doing this is to use Pokemon that have very different viable movesets and try to trick your opponent into thinking you're playing one moveset when you're actually playing another. So in this respect Pokemon resembles poker more than it does chess.