Sleeping Beauty Loses Her Magic Psychic Powers

An agent, Sleeping Beauty, in a game with self-locating uncertainty (i.e. one play of the game visits the same information set multiple times, as in the paradox of the absentminded driver) must select a behavioural strategy that is self-ratifying: a best-response to the belief that her other instances do likewise. When there are multiple such fixed points, the standard treatment of Aumann et al. (1997) assumes that all instances of the agent can simply coordinate. I drop this assumption (supposing that Sleeping Beauty 'lose her magic psychic powers' ), and study said agent iteratively reasoning her way to an equilibrium selection instead. Which strategy other instances select can be seen as ambiguous, so I model Beauty's choice via a response function that encodes her response to ambiguity, and a procedure that describes in what manner she iterates the application of this response function. I consider two response functions, 'Bayesian' and EU-maximin, and two procedures, replacement and accumulation, and characterise in which cases the iterative reasoning converges. For either response function, accumulation always converges, but replacement does so if and only if there are no cycles of length weakly greater than two on a particular finite functional digraph I call the support digraph.