Have any studies been done that demonstrate people (not game theorists) actually using mixed Nash equilibrium as their strategy in a game?
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According to this(article about mixed equilibrium strategies), I think penalty kicks between two soccer teams use mixed Nash equilibrium strategies.
There have been lots of studies on this sort of thing, with different results. It depends a lot on cultural context. You might look at "A Beautiful Math" by Tom Siegfried
My preliminary research backs up my intuition: it depends. Specifically, it depends on whether there is sufficient common knowledge to foster a gradient-like movement to mixed-equilibria.
Learning a Mixed Strategy Equilibrium in the Laboratory by Bloomfeld (1994)
An Experimental Study of Information and Mixed-Strategy Play in the Three-Person Matching-Pennies Game by McCabe, Mukherji and Runkle (2000)
How do People Play Against Nash Opponents? by Shachat, Swarthout, Wei (2011)