Kripke Semantics of the Perfectly Transparent Equilibrium

Abstract: The Perfectly Transparent Equilibrium is algorithmically defined, for any game in normal form with perfect information and no ties, as the iterated deletion of non-individually-rational strategy profiles until at most one remains. In this paper, we characterize the Perfectly Transparent Equilibrium with adapted Kripke models having necessary rationality, necessary knowledge of strategies as well as eventual logical omniscience. Eventual logical omniscience is introduced as a weaker version of perfect logical omniscience, with logical omniscience being quantized and fading away counterfactually. It is the price to pay for necessary factual omniscience and necessary rationality: we conjecture that epistemic omniscience, logical omniscience and necessary rationality form an impossibility triangle. We consider multimodal classes of Kripke structures, with respect to agents, but also in the sense that we have both epistemic and logical accessibility relations. Knowledge is defined in terms of the former, while necessity is defined in terms of the latter. Lewisian closest-state functions, which are not restricted to unilateral deviations, model counterfactuals. We use impossible possible worlds `a la Rantala to model that some strategy profiles cannot possibly be reached in some situations. Eventual logical omniscience is then bootstrapped with the agents' considering that, at logically possible, but non-normal worlds `a la Kripke, any world is logically accessible and thus any deviation of strategy is possible. As in known in literature, under rationality and knowledge of strategies, these worlds characterize individual rationality. Then, in normal worlds, higher levels of logical omniscience characterize higher levels of individual rationality, and a high-enough level of logical omniscience characterizes, when it exists, the Perfectly Transparent Equilibrium.