An Introductory Analysis of Competitive Pokemon Using Game Theory
At the heart of the Pokemon role-playing games is a turn-based battling feature in which two players try to reduce the health points (HP) of all six of the opponent’s Pokemon to zero, one Pokemon at a time. This competitive aspect of Pokemon assumes the players have spent an unlimited amount of time in preparing their six Pokemon, so the matchup is isolated from factors such as the level of the Pokemon, the evolutionary stage of the Pokemon and the training values of the Pokemon; each Pokemon is trained beforehand to its utmost potential.
Before each turn, each player chooses a command for their Pokemon in play, which can be to attack the opposing Pokemon or switch it for another Pokemon (but only one may be chosen). If player 2 chooses to switch and player 1 chooses to attack, the switch always happens before the attack. Some Pokemon have an advantage over others; for this article we shall assume that advantage is solely type-based: Fire beats Grass, Grass beats Water, and Water beats Fire. It is very often the case that the two Pokemon in play are of different types, and so one has an advantage.
Assume that player 1 has a Fire-type Pokemon in play and player 2 has a Grass-type Pokemon in play. Player 1 seems to be at an advantage, and player 2 seems to be inclined to switch. Let us assign an advantage of this kind a value 10. If player 1 attacks the opposing disadvantaged Pokemon during the turn, he gains value 20 as a Fire-type attack would be “super effective” against a Grass-type Pokemon. However, if player 2 chooses to switch to a Water-type Pokemon, player 1’s gain is only 5, since Fire-type is “not very effective” against Water-type, but player 2 gains the advantage of 10, because player 2 would then have the upper hand in typing. Knowing this, player 1 can choose not to attack and instead also switch to maintain the upper hand in typing, yet he runs the risk of player 2 choosing not to switch. We can model this scenario in a gains table:
| Player 2 chooses attack | Player 2 chooses switch | |
| Player 1 chooses attack | (20, 5) | (5, 10) |
| Player 1 chooses switch | (0, 20) | (0, 0) |
There is no pure strategy Nash equilibrium in this case; nor is there a dominant strategy. If player 1 chooses attack, 2’s best strategy is to switch. If player 2 chooses switch, 1’s best strategy is to switch. If player 1 chooses switch, 2’s best strategy is to attack. If player 2 chooses attack, 1’s best strategy is to attack.
We can, however, calculate the mixed strategy Nash Equilibrium. Let p be the probability that player 1 chooses attack and (1 – p) be the probability he chooses switch. Let q represent these probabilities for player 2. Then we have:
| 20q + 5(1 – q) = 10(1 – q) |
| 5p + 20(1 – p) = 10p |
Solving, we get that if player 1 chooses to attack with a likelihood of p = 4/5 and player 2 chooses to attack with a likelihood of q = 1/5. The mixed strategy Nash equilibrium implies that if player 1 believes player 2 to act with such a probability, then player 1 would also act with such a probability, and vice versa. Under this likelihood neither player would have greater gain over several turns. The fact that there exists only a mixed strategy equilibrium and not a pure one is a key reason that many entry level competitive players consider Pokemon a game of probability, where choosing what to play on a turn rests on where the Nash equilibrium lies, simply that there is a much larger set of factors that affect the gain values than provided in this example, including attacks that have special effects, attacks of different types, which Pokemon attacks first in one turn, and much more.
Consulted articles:
Kill or Be Killed – A UU Offense Analysis, Smogon: http://www.smogon.com/smog/issue10/uu.
Attacking Types, Smogon: http://www.smogon.com/smog/issue3/attacking_types.