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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?

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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.

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  • $\begingroup$ Even in an iterated PD where you might try a Tit for Tat strategy? $\endgroup$
    – bigbucky
    Commented Aug 4, 2020 at 17:47
  • $\begingroup$ If there is a fixed number of iterations known to both players then by induction it is best to always defect. This is because both players will defect on the last iteration, so because both players know they will defect on the last, they will also both defect on the second to last iteration because there is nothing to lose. This reasoning can be continued and they will both always defect. $\endgroup$
    – cha21
    Commented Aug 4, 2020 at 17:51

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