In a simultaneous move game; if a strategy is never the best response, is it dominated?

Question

For example, consider the strategic form game:

In this case, P2 never plays $R$ and P1 never plays $B$. $R$ is dominated by $C$ and $B$ is dominated by playing $T,M$ with respective probabilities $\frac{4}{5},\frac{1}{5}$. Is it always the case that if a strategy is never the best response in a simultaneous move game, then it is dominated?

Answer 1

In a finite game in strategic form, a strategy of a player is never a best response if and only if it is strictly dominated; see Lemma 60.1 in Osborne and Rubinstein, A course in Game Theory, 1994.

This lemma implies that an action that is weakly dominated - but not strictly dominated - is a best response to some belief.

(The question does not make clear which notion of dominance is intended.)

Answer 2

It could be that although a certain pure strategy $S$ of player $1$ is not dominated by any of the others, because it would produce the best payoff for one particular pure strategy $T$ of player $2$, strategy $T$ is dominated by other strategies of player $2$ and thus should never be played. Perhaps for all of player $2$'s pure strategies except $T$, strategy $S$ is worse than strategy $S'$. In this situation an optimal strategy for player $1$ will never include $S$.