Gambling using a finite state machine

Abstract

Sequential gambling schemes in which the amount wagered on the future outcome is determined by a finite state (FS) machine are defined and analyzed. It is assumed that the FS machine determines the fraction of the capital wagered at each time instance i on the outcome at the next time instance, i+1, and that wagers are paid at even odds. The maximal capital achieved by any FS machine is found and its dependence on an empirical entropy measure, H/sup FS/(x), defined as the finite state complexity of x, is shown. A specific gambling scheme is then proposed based on the Lempel-Ziv method for universal compression. The capital gained by this method is found and it is observed that, asymptotically, its exponential growth rate dominates the experimental growth rate achieved by gambling using any FS machine. Furthermore, this specific scheme suggests a class of gambling methods, based on a class of variable-to-variable length lossless compression methods, in which the capital is doubled for every bit compressed. These results emphasize the relation between gambling and data compression. >