A family of self-organizing systems
Cornell University, Center for Applied Mathematics, 1966 - Computers - 106 pages
An investigation was made of a class of adaptive systems and the systems' behaviors as game-playing machines. Each member of the class is described by a set of parameters that specifies its reenforcement mechanism. In general, such a mechanism tends to increase successful strategies' probabilities of occurrence. However, the parameters must be carefully selected if the adaptive system's probability of winning is to approach one. The paper first develops a class of urn models, described by the same parameters; and shows that each urn model behaves very much like a corresponding adaptive system. The familiar urns of Polya and Barnard Friedman are members of this class. Other members exhibit much more interesting behaviors. The paper analyzes the urn models and proves a sufficient condition for convergence to one, with probability one, of the right-ball ratio. It exhibits numerical results showing that, for practical applications, the condition is also necessary. Finally, it analyzes the glitch phenomenon. (Author).
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absolute moments additional card labeled appearing within absolute asymptotic backhand diagonal ball is drawn ball will converge card is drawn card is returned compute conditional probability conjecture Cornell University cup participating cup seven cup two receives defined distribution functions drawing a right drawn card drawn from cup end-game Equation 26 glitch inequality inherently positive integer intransigent point large values last chip Last One Loses lattice points learning machine Lemma Three loss occurs monotone decreasing non-negative number of balls number of right optimal strategy path player B lost Player B removes player B won positive numbers positive reinforcement probability measure probability of drawing probability of winning random variable receive negative reinforcements result rewritten right ball right card right draws sample space sets of parameters seven receive step function sufficiently large Theorem three chips total number transition probabilities unit step urn model winning converges wrong balls wrong card wrong draws