Game Theory rationalizable strategies that are all the same

Question

Suppose player 1 has strategies u, m, and d, and player 2 has strategies A, B, and C. The strategies A, B and C are all identical (payoff equivalent). Can any of them be eliminated using the rationalizable strategies method? The only other info for A is that u is a strictly dominated strategy.

Answer 1

1) For two player games, the set of rationalizable strategies coincides with the set of strategies that survive the process of iterative elimination of strictly dominated strategies. Because A, B, and C are payoff equivalent and there are no other strategies for player 2 besides A, B, and C, none of these strategies is strictly dominated so all strategies of player 2 are rationalizable.

2) Even if all strategies are rationalizable, often one eliminates strategies that are payoff equivalent (leaving just one representative of equivalence the class$^\dagger$) in order to simplify the game.

$\dagger\quad$ The relation $s\in S_i \sim t \in S_i \Leftrightarrow \forall s_{-i}\in S_{-i},\, u_i(s,s_{-i})=u_i(t,s_{-i})$ defines equivalent classes in $S_i$ and we work with the quotient $S_i/\sim$ space instead of $S_i$.