What is the point of considering only pure strategies in a game? How could you restrict people from thinking about mixed strategy?
If in equilibrium, a player &quot;chooses a mixed strategy&quot; that plays $H$ and $T$ with positive probability, $H$, and $T$ must be both optimal choices. It is a standard result that for a (subjective or objective) expected utility maximizer, randomizing can only be optimal if it is over pure optimal choices. This is a direct consequence of expected utilities being linear in probabilities. So the problem you mentioned can never occur.
So expected utility maximizers do not ever have a strict incentive to randomize, which raises the question of how one can interpret Nash equilibria in mixed strategies. One popular interpretation is that a player's mixed strategy is really representing a shared probabilistic belief other players have over the choices of that player. One can then define Nash equilibrium as a consistency condition on beliefs, though it is not clear why this condition should obtain in practice. Oner idea, Harsanyi's purification idea is that with a certain probability the player has private information that makes $H$ or $T$ uniquely optimal, but for the other players it will look like choosing $H$
and $T$ randomly. Especially in zero-sum games, there is also the idea that players deliberately use randomization devices such as coins so that their plans cannot be found out. Lastly, there is the &quot;mass action&quot; interpretation that can already be found in Nash's thesis. According to this interpretation, you are randomly matched with players from a huge population that play a fixed pure strategy, and the mixing probabilities represent the population fractions playing each pure strategy.