I haven't read up on game theory in a long time, does "best strategy to follow" mean "the strategy that maximizes my expected payoff for all the strategies my o

I haven't read up on game theory in a long time, does "best strategy to follow" mean "the strategy that maximizes my expected payoff for all the strategies my opponent could employ"? So, an explicit counterexample would be my opponent picking a strategy of only re-throwing above .25. His expectation is then .25 * (0 + .25)/2 + .75 * (0 + 1)/2 ~= .40, so I should not rethrow if I get above .40 and below .50, even though it would raise my own expectation. Am I thinking about that correctly?