The dynamics of decision making when probabilities are vaguely specified

Abstract

Consider a multi-trial game with the goal of maximizing a quantity Q(N). At each trial N, the player doubles the accumulated quantity, unless the trial number is Y, in which case all is lost and the game ends. The expected quantity for the next trial will favor continuing play, as long as the probability that the next trial is Y is less than one half. Y is vaguely specified (e.g., someone is asked to fill a sheet of paper with digits, which are then permuted to produce Y). Conditional on reaching trial N, we argue that the probability that the next trial is Y is extremely small (much less than one half), and that this holds for any N. Thus, single trial reasoning recommends one should always play, but this guarantees eventual ruin in the game. It is necessary to stop, but how can a decision to stop on N be justified, and how can N be chosen? The paradox and the conflict between what seem to be two equally plausible lines of reasoning are caused by the vagueness in the specification of the critical trial Y. Many everyday reasoning situations involve analogous situations of vagueness, in specifying probabilities, values, and/or alternatives, whether in the context of sequential decisions or single decisions. We present a computational scheme for addressing the problem of vagueness in the above game, based on quantum probability theory. The key aspect of our proposal is the idea that the range of stopping rules can be represented as a superposition state, in which the player cannot be assumed to believe in any specific stopping rule. This scheme reveals certain interesting properties, regarding the dynamics of when to stop to play.