## A family of self-organizing systemsAn 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 adaptive system additional card labeled appearing within absolute asymptotic bability backhand diagonal ball is drawn ball will converge balls are added card is returned compute conditional probability conjecture contains some right cup participating cup seven cup two receives cup two's defined drawing a right drawn card drawn from cup end-game Enter Equation FAMILY OF SELF-ORGANIZING Game-Playing Machine glitch inequality integer intransigent point large values Last One Loses lattice points Lemma Three Markov Process monotone decreasing Note number of balls number of right optimal strategy pile player B loses Player B removes player B won positive numbers positive reinforcement probability measure probability of drawing probability of winning random variable random walk receive negative reinforcements rewritten right ball right card sample space SELF-ORGANIZING SYSTEMS sets of parameters SPONSORING MILITARY sufficiently large Theorem total number unit step urn contains urns of Polya winning converges wrong balls wrong card zero