A functional equation analysis of two learning models

One-absorbing barrier random walks arising from Luce's nonlinear beta model for learning and a linear commuting-operator model (called the alpha model) are considered. Functional equations for various statistics are derived from the branching processes defined by the two models. Solutions to general functional equations, satisfied by statistics of the alpha and beta models, are obtained. The methods presented have application to other learning models.Abstracted from portions of the author's doctoral dissertation, University of Pennsylvania, June 1960. The author is indebted to Robert R. Bush, his dissertation supervisor for the valuable help and encouragement received from him and to R. Duncan Luce for many helpful discussions and for partial support from an NSF grant.The author is grateful to J. G. Brainerd, S. Gorn, and C. N. Weygandt of the Moore School, and N. F. Finkelstein, D. Parkhill and A. A. Wolf of General Dynamics for their encouragement.     

The asymptotic distribution for the two-absorbing-barrier beta model

For the two-absorbing-barrier specialization of Luce's beta learning model, the asymptotic distribution of the response probability has all its density atp = 0 andp = 1. The functional equation for the amount of the density atp = 1 is investigated in this paper.Abstracted from a portion of the author's doctoral dissertation, University of Pennsylvania, June 1960. The author is indebted to Prof. B. Epstein and to Prof. Robert R. Bush, his dissertation supervisor, for the valuable help and encouragement received from them.The author is grateful to the Moore School for the support extended to him during his doctoral studies. He also wishes to thank D. Parkhill and N. Finkelstein of General Dynamics for their encouragement of his work.