This is the problem.

For this question, I believe Albus has 3 strategies: N, E, and S, and Minerva has 8: aaa, aab, aba, abb, baa, bab, bba, and bbb. I've also found the rollback equilibrium to be the case in which Albus chooses N and Minerva chooses b. However, I'm not sure how to represent this equilibrium (equilibria?).

Do we say that the unique equilibrium is simply (N, b), since those are the choices made by the players, or do we say there are 4 equilibria: (N, baa), (N, bab), (N, bba), (N, bbb) since Minerva's choices are in reality a set of strategies, all of which result in her choosing b in response to Albus playing N?


You're correct on the definition of the strategies: Albus' strategy set is $S_{A} = \big\{N,E,S\big\}$, and Minerva's is $S_{M} = \big\{a, b\big\} \times \big\{a, b\big\} \times \big\{a, b\big\}$.

Remember, a strategy is a contingent plan that prescribes an action at every possible node at which a player plays.

Then, the unique equilibrium (via Backward Induction) is $(N, b, a, b)$. The others you mention are not equilibria, because they involve incredible threats, e.g. if Minerva found herself at the decision node following a choice of $E$ by Albus, she would never choose $b$.


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