Blackwells Demon: Postdiction and Prediction in Random Walks

Abstract

Maxwells Demon is a mythical being, first described by the physicist James Clerk Maxwell (although named Maxwells Demon by Lord Kelvin). Maxwell used it in a thought experiment to potentially violate the Second Law of Thermodynamics by exploiting inhomogeneities existing in a statistically homogeneous system. Blackwells Demon, making (as far as is known) its first appearance in this paper, illustrates a counterintuitive situation occurring in a random walk variation of the Two Envelope problem[1], that it is possible under restrictive conditions to predict with success probability > 1/2 the direction of a random walk generated by the flip of a fair coin. Like Maxwells Demon, Blackwells Demon operates by exploiting inhomogeneities that exist in a statistically homogeneous system. Maxwells Demon achieves its results by knowing when a molecule is moving rapidly and when it is not. Blackwells Demon achieves its results by knowing when a prediction strategy is successful and when it is not. At the time Maxwell proposed his Demon, it confronted a technological Everest, the ability to open and close a gate permitting the passage of a single molecule, and the ability to gauge the speed of an approaching molecule. Blackwells Demon merely has to turn a light on and off, conduct visual observations and keep simple statistical records. It should be noted that the analysis in this paper does not demonstrate the ability to predict the flip of a fair coin ab initio with success probability > 1/2, as it is necessary to embed the fair coin in an environment of some complexity in order to achieve this result.