What is the point of considering only pure strategies in a game? How could you restrict people from thinking about mixed strategy? In an experimental setting, how could you effectively incentivize the subjects to not to adopt mixed strategy?
I would like to re-emphasize that the question in concern is &quot;how to prevent people from using mixed strategy&quot; such that only pure strategies are adopted. The mixed strategies must be theoretically adoptable, and we have a mechanism to force people to think about pure strategies only. In a non-repeated game, if a person plays strategy &quot;H&quot;, in general you don't know if it is pure strategy &quot;H&quot; or a mixed strategy with positive probability on &quot;H&quot;. The current answers are very useful and well-prepared; however, what I am always looking for is a proven (either theoretically or experimentally) method that constrains the choice set from a mixture space to a doublet.
Technically, in a game theory experiment, the set of alternatives is a mixture set. I want to restrict the set of alternative to two objects, $\{H, T\}$, only.
Of course, you could just post a title of a paper which includes an incentive mechanism or an experimental design. This can be a perfect answer despite of its length. Opinions are welcome but opinions are not answers.

Let's consider a one-period game where the first player choose $H$ or $T$. The game satisfies the following two conditions:

The equilibrium is unique at $\frac{2}{3} H+\frac 1 3 T$.

If the player is restricted from choosing mixed strategy, $T$ becomes the optimal choice for player 1.


I've chatted with a few people and they all believe that it makes sense to consider those two cases separately; by &quot;separately&quot; they mean that, a game problem restricting the player from choosing mixed strategy also make sense, we can compare $H$ and $T$ in isolation of those mixed strategies. What is the philosophy behind this?
I, on the other hand, believe that the latter case doesn't make sense in real life: one cannot just abandon the mixed strategy completely. In an experimental setting, how could you prevent the players from adopting a mixed strategy?
One awkward way to impose this restriction, I think, is to instruct the player that it is a super game repeated for 10 times (or even life-time); you can only choose the same $H$ or $T$ for the lifetime and never change your choice. But this way, people can still think about mixed strategy.

Clarification: I am not advocating that people must use mixed strategy only. I am just saying that, I cannot find a good experiment that restricts people from including mixed strategies in their menu. Similarly, I cannot find a good experiment that restricts people from including pure strategies in their choice sets. So I think, in analysis, we must think about all strategies together, and it is pointless to consider pure or mixed strategies only.
PS: Assume the player is rational, as this is economics SE.
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.