What happens if the players in a prisoner’s dilemma or stag hunt game don’t always have control over their choices?
Instead of deciding to cooperate or defect the players have to draw from a deck. There are an equal number of cooperate and defect cards and each player picks two. They must each play one card. 50% of the time (they have both cards) they can make their own decision about what to do. But the other times it is not their choice since they have two of the same cards.
Does this do anything to the optimal strategy? Does it just take longer to come to equilibrium as you try to guess if they really meant to defect or they were forced to?
The strategy does not change. The key point is from either player's perspective it doesn't matter if the other player intended to defect or was forced to, it is always better to defect regardless of the choice of the other player or how they made the desision.