Perfect Prediction in Normal Form: Superrational Thinking Extended to Non-Symmetric Games
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This paper introduces a new solution concept for non-cooperative games in normal form with no ties and with pure strategies: the Perfectly Transparent Equilibrium. The players are assumed to be Perfect Predictors, in the sense that their predictions are correct in all possible worlds. Concretely, this means that a player's decision is perfectly correlated with its anticipation by other players, unlike in Nash equilibra where the decisions are made independently. The equilibrium, when it exists, is unique and is Pareto optimal. This equilibrium can be seen as the natural normal-form counterpart of the Perfect Prediction Equilibrium, hitherto defined on games in extensive form. The core difference between the reasonings in extensive form and in normal form is that the decision and its prediction are spacelike-separated instead of being timelike-separated. Like in the PPE, the equilibrium is computed by iterated elimination of preempted strategy profiles until at most one remains. An equilibrium is a strategy profile that is immune against its common knowledge, in the sense that players have no interest in deviating from their strategies. The equilibrium can also be seen as a natural extension of Hofstadter's superrationality to non-symmetric games. Indeed, on symmetric games, the equilibrium, when it exists, perfectly coincides with the Hofstadter equillibrium. This paper defines the Perfectly Transparent Equilibrium and, in the case that it exists, proves its uniqueness, its Pareto-optimality, and that it coincides with the Hofstadter equillibrium on symmetric games. We also relate preemption to individual rationality and give hints about further research to relate this equilibrium to other non-Nashian solution concepts such as the Perfect Prediction Equilibrium in extensive form, Halpern's and Pass's minimax-rationalizability and Shiffrin's joint-selfish-rational equilibrium.
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