Consider the following two player game. Each player picks a Pokemon type, without revealing it to the other player. Each player then "deals damage" to the other player as if we are using the Pokemon advantage rules. So if your type is super effective on the opponents type, you will deal 200 damage, it is not very effective, it will deal 50 damage, if it has no effect, it will deal 0 damage, otherwise it will do 100 damage. This is played 10 times, and the player that has dealt the most damage wins. What is the optimal strategy?
As the amount of damage does not matter, only the fact whether or not you dealt more damage than the opponent, we can consider this a zero-sum game were each player only remembers the amount of damage that was dealt more than the enemy.
The type advantage table is this:| 📦 | 🔥 | 💧 | ⚡ | 🌿 | ❄️ | 🥊 | ☠️ | ⛰️ | 🐦 | 👁️ | 🐞 | 🗿 | 👻 | 🐲 | 🌙 | ⚙️ | ✨ | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 📦 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 0.5 | 0.0 | 1.0 | 1.0 | 0.5 | 1.0 |
| 🔥 | 1.0 | 0.5 | 0.5 | 1.0 | 2.0 | 2.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 2.0 | 0.5 | 1.0 | 0.5 | 1.0 | 2.0 | 1.0 |
| 💧 | 1.0 | 2.0 | 0.5 | 1.0 | 0.5 | 1.0 | 1.0 | 1.0 | 2.0 | 1.0 | 1.0 | 1.0 | 2.0 | 1.0 | 0.5 | 1.0 | 1.0 | 1.0 |
| ⚡ | 1.0 | 1.0 | 2.0 | 0.5 | 0.5 | 1.0 | 1.0 | 1.0 | 0.0 | 2.0 | 1.0 | 1.0 | 1.0 | 1.0 | 0.5 | 1.0 | 1.0 | 1.0 |
| 🌿 | 1.0 | 0.5 | 2.0 | 1.0 | 0.5 | 1.0 | 1.0 | 0.5 | 2.0 | 0.5 | 1.0 | 0.5 | 2.0 | 1.0 | 0.5 | 1.0 | 0.5 | 1.0 |
| ❄️ | 1.0 | 0.5 | 0.5 | 1.0 | 2.0 | 0.5 | 1.0 | 1.0 | 2.0 | 2.0 | 1.0 | 1.0 | 1.0 | 1.0 | 2.0 | 1.0 | 0.5 | 1.0 |
| 🥊 | 2.0 | 1.0 | 1.0 | 1.0 | 1.0 | 2.0 | 1.0 | 0.5 | 1.0 | 0.5 | 0.5 | 0.5 | 2.0 | 0.0 | 1.0 | 2.0 | 2.0 | 0.5 |
| ☠️ | 1.0 | 1.0 | 1.0 | 1.0 | 2.0 | 1.0 | 1.0 | 0.5 | 0.5 | 1.0 | 1.0 | 1.0 | 0.5 | 0.5 | 1.0 | 1.0 | 0.0 | 2.0 |
| ⛰️ | 1.0 | 2.0 | 1.0 | 2.0 | 0.5 | 1.0 | 1.0 | 2.0 | 1.0 | 0.0 | 1.0 | 0.5 | 2.0 | 1.0 | 1.0 | 1.0 | 2.0 | 1.0 |
| 🐦 | 1.0 | 1.0 | 1.0 | 0.5 | 2.0 | 1.0 | 2.0 | 1.0 | 1.0 | 1.0 | 1.0 | 2.0 | 0.5 | 1.0 | 1.0 | 1.0 | 0.5 | 1.0 |
| 👁️ | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 2.0 | 2.0 | 1.0 | 1.0 | 0.5 | 1.0 | 1.0 | 1.0 | 1.0 | 0.0 | 0.5 | 1.0 |
| 🐞 | 1.0 | 0.5 | 1.0 | 1.0 | 2.0 | 1.0 | 0.5 | 0.5 | 1.0 | 0.5 | 2.0 | 1.0 | 1.0 | 0.5 | 1.0 | 2.0 | 0.5 | 0.5 |
| 🗿 | 1.0 | 2.0 | 1.0 | 1.0 | 1.0 | 2.0 | 0.5 | 1.0 | 0.5 | 2.0 | 1.0 | 2.0 | 1.0 | 1.0 | 1.0 | 1.0 | 0.5 | 1.0 |
| 👻 | 0.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 2.0 | 1.0 | 1.0 | 2.0 | 1.0 | 0.5 | 1.0 | 1.0 |
| 🐲 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 2.0 | 1.0 | 0.5 | 0.0 |
| 🌙 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 0.5 | 1.0 | 1.0 | 1.0 | 2.0 | 1.0 | 1.0 | 2.0 | 1.0 | 0.5 | 1.0 | 0.5 |
| ⚙️ | 1.0 | 0.5 | 0.5 | 0.5 | 1.0 | 2.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 2.0 | 1.0 | 1.0 | 1.0 | 0.5 | 2.0 |
| ✨ | 1.0 | 0.5 | 1.0 | 1.0 | 1.0 | 1.0 | 2.0 | 0.5 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 2.0 | 2.0 | 0.5 | 1.0 |
So the payout matrix is:
| 📦 | 🔥 | 💧 | ⚡ | 🌿 | ❄️ | 🥊 | ☠️ | ⛰️ | 🐦 | 👁️ | 🐞 | 🗿 | 👻 | 🐲 | 🌙 | ⚙️ | ✨ | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 📦 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | -1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | -0.5 | 0.0 | 0.0 | 0.0 | -0.5 | 0.0 |
| 🔥 | 0.0 | 0.0 | -1.5 | 0.0 | 1.5 | 1.5 | 0.0 | 0.0 | -1.0 | 0.0 | 0.0 | 1.5 | -1.5 | 0.0 | -0.5 | 0.0 | 1.5 | 0.5 |
| 💧 | 0.0 | 1.5 | 0.0 | -1.0 | -1.5 | 0.5 | 0.0 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | -0.5 | 0.0 | 0.5 | 0.0 |
| ⚡ | 0.0 | 0.0 | 1.0 | 0.0 | -0.5 | 0.0 | 0.0 | 0.0 | -2.0 | 1.5 | 0.0 | 0.0 | 0.0 | 0.0 | -0.5 | 0.0 | 0.5 | 0.0 |
| 🌿 | 0.0 | -1.5 | 1.5 | 0.5 | 0.0 | -1.0 | 0.0 | -1.5 | 1.5 | -1.5 | 0.0 | -1.5 | 1.0 | 0.0 | -0.5 | 0.0 | -0.5 | 0.0 |
| ❄️ | 0.0 | -1.5 | -0.5 | 0.0 | 1.0 | 0.0 | -1.0 | 0.0 | 1.0 | 1.0 | 0.0 | 0.0 | -1.0 | 0.0 | 1.0 | 0.0 | -1.5 | 0.0 |
| 🥊 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 | -0.5 | 0.0 | -1.5 | -1.5 | 0.0 | 1.5 | -1.0 | 0.0 | 1.5 | 1.0 | -1.5 |
| ☠️ | 0.0 | 0.0 | 0.0 | 0.0 | 1.5 | 0.0 | 0.5 | 0.0 | -1.5 | 0.0 | -1.0 | 0.5 | -0.5 | -0.5 | 0.0 | 0.0 | -1.0 | 1.5 |
| ⛰️ | 0.0 | 1.0 | -1.0 | 2.0 | -1.5 | -1.0 | 0.0 | 1.5 | 0.0 | -1.0 | 0.0 | -0.5 | 1.5 | 0.0 | 0.0 | 0.0 | 1.0 | 0.0 |
| 🐦 | 0.0 | 0.0 | 0.0 | -1.5 | 1.5 | -1.0 | 1.5 | 0.0 | 1.0 | 0.0 | 0.0 | 1.5 | -1.5 | 0.0 | 0.0 | 0.0 | -0.5 | 0.0 |
| 👁️ | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.5 | 1.0 | 0.0 | 0.0 | 0.0 | -1.0 | 0.0 | -1.0 | 0.0 | -2.0 | -0.5 | 0.0 |
| 🐞 | 0.0 | -1.5 | 0.0 | 0.0 | 1.5 | 0.0 | 0.0 | -0.5 | 0.5 | -1.5 | 1.0 | 0.0 | -1.0 | -0.5 | 0.0 | 1.0 | -0.5 | -0.5 |
| 🗿 | 0.5 | 1.5 | -1.0 | 0.0 | -1.0 | 1.0 | -1.5 | 0.5 | -1.5 | 1.5 | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 | -1.5 | 0.0 |
| 👻 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 1.0 | 0.5 | 0.0 | 0.0 | 1.0 | 0.5 | 0.0 | 0.0 | 0.0 | -1.5 | 0.0 | 0.0 |
| 🐲 | 0.0 | 0.5 | 0.5 | 0.5 | 0.5 | -1.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | -0.5 | -2.0 |
| 🌙 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | -1.5 | 0.0 | 0.0 | 0.0 | 2.0 | -1.0 | 0.0 | 1.5 | 0.0 | 0.0 | 0.0 | -1.5 |
| ⚙️ | 0.5 | -1.5 | -0.5 | -0.5 | 0.5 | 1.5 | -1.0 | 1.0 | -1.0 | 0.5 | 0.5 | 0.5 | 1.5 | 0.0 | 0.5 | 0.0 | 0.0 | 1.5 |
| ✨ | 0.0 | -0.5 | 0.0 | 0.0 | 0.0 | 0.0 | 1.5 | -1.5 | 0.0 | 0.0 | 0.0 | 0.5 | 0.0 | 0.0 | 2.0 | 1.5 | -1.5 | 0.0 |
Using the technique described here we can find a Mixed strategy that solves this game.
The mixed strategy we use looks like this:| pick | deal | recieve | overflow | |
|---|---|---|---|---|
| Water 💧 | 0.1640 | 0.9932 | 0.9932 | 0.0000 |
| Dragon 🐲 | 0.1040 | 1.0026 | 1.0026 | 0.0000 |
| Grass 🌿 | 0.0982 | 1.0193 | 1.0193 | -0.0000 |
| Electric ⚡ | 0.0907 | 0.9780 | 0.9780 | 0.0000 |
| Fire 🔥 | 0.0898 | 1.0569 | 1.0569 | 0.0000 |
| Ground ⛰️ | 0.0865 | 1.2078 | 1.2078 | 0.0000 |
| Steel ⚙️ | 0.0793 | 0.9080 | 0.9080 | 0.0000 |
| Ghost 👻 | 0.0617 | 1.0544 | 1.0544 | 0.0000 |
| Fairy ✨ | 0.0617 | 1.0121 | 1.0121 | 0.0000 |
| Ice ❄️ | 0.0582 | 1.1400 | 1.1400 | -0.0000 |
| Flying 🐦 | 0.0470 | 1.0132 | 1.0132 | -0.0000 |
| Poison ☠️ | 0.0441 | 0.9845 | 0.9845 | -0.0000 |
| Dark 🌙 | 0.0147 | 1.0235 | 1.0235 | 0.0000 |
| Normal 📦 | 0.0000 | 0.8986 | 0.9383 | -0.0397 |
| Bug 🐞 | 0.0000 | 0.9211 | 1.0444 | -0.1233 |
| Psychic 👁️ | 0.0000 | 0.9897 | 1.0764 | -0.0867 |
| Fighting 🥊 | 0.0000 | 1.0141 | 1.1014 | -0.0873 |
| Rock 🗿 | 0.0000 | 1.1121 | 1.3377 | -0.2256 |
In this table, "pick" decides the probability of each pure strategy of picking that specific type, "deal" is the expected amount of damage that the user deals if this type is used against this strategy, "receive" is the expected damage that will be received from this strategy, and overflow the difference between the two. Note that for a Nash Equilibrium, the overflow value has to be negative, because if it was positive increasing the probability of this strategy would create a strategy that would beat this. Analogous, for each type with a nonzero probability, the overflow cannot be negative, otherwise removing this type from the strategy would result in a new mixed strategy that can defeat this strategy.
putting the non zero values in a pie-chart looks like this: