Relationship between Difficulty of Weakly and Strongly Solving a Game

Question

For a given game, is there any relationship between the computational complexity of weakly solving the game and strongly solving it? I ask because a weakly, but not strongly solved game with simple strategies to guarantee at least a draw could still be interesting to play if the players place little value on a draw, and do not know how to win if their opponent is also trying to achieve more than just a draw. However, if simple strategies weakly solving a game guaranteed simple strategies strongly solving it, then this point would be moot. So, if for a given game, as some parameter scales up we have a weak solution of a particular complexity class, can we bound the complexity class of a strong solution?