Using the Cover Guessing Game Strategy to Predict the Results of Stochastic Processes

Abstract

Cover originally proposed a game in which Bob wrote two numbers on pieces of paper and allowed Alice to pick one and look at it. Cover then showed, as Blackwell is thought to have shown previously, that Alice has a strategy involving the selection of a random number which enables her to guess correctly with a probability greater than 0.5 which of the two numbers was larger. Central to Covers reasoning was that Alice could have picked either number to look at with equal probability. In this paper, we investigate the situation in which Bob obtains the two numbers as the result of an experiment based on a probability distribution, such as the spin of a roulette wheel. We then modify this to envision Alice and Bob conducting separate experiments based on probability distributions. Alice conducts one trial of her experiment and uses Covers strategy to guess whether the result from her experiment will be larger or smaller than the result from Bobs experiment whenever he chooses to conduct a single trial of it. The following two results are among those in the paper. (1) If the two numbers are observations of the same experiment, Alice merely has to observe one trial and apply the Cover strategy to have a probability greater than 0.5 of guessing the larger value, even if the second trial has not yet been conducted (2) There are many nontrivial examples where Alice and Bob conduct different experiments, and Alice can observe one trial of her experiment and apply the Cover strategy to have a probability greater than 0.5 of guessing the larger value, even if many of the specifics of the other experiment have not yet been determined and a trial of it has not yet been conducted.