There are 18 different types in pokemon with different strengths and weaknesses like rock-paper-scissors. For example, Fire beats Grass, Grass beats Water, and Water beats Grass. My goal is to determine the best type of pokemon (ignoring stats, special abilities, move sets, turn order, etc.) In this simplified game of pokemon, you pick one pokemon type and the pokemon will use that pokemon type attack as well (if you pick fire, then you get a generic fire pokemon that only uses a generic fire move and likewise for your opponent for their chosen type).
I have 2 approaches to the problem.
Make a matrix such that each column is a type and each entry in that column is probability that it loses to a different type (as indicated by row, and each probability is either 0 or uniform). Find the eigenvector for this matix. Whichever type has the higher probability is the best type.
Solve the nash equilibrium for this game. Find where the opponent is indifferent to the probability distribution to how you pick your type. This would be a simultaneous game.
I tried both approaches but got different results. Why?