Dynamic paired-comparison scaling.
type: Item
instance of: scholarly article
title: Dynamic paired-comparison scaling.
zbMATH Open document ID: 1123.62311
DOI: 10.1016/0022-2496(92)90036-7
author: William H. Batchelder
author: N. J. Bershad
author: R. Simpson
published in: Journal of Mathematical Psychology
publication date: 1992-09-27T00:00:00Z
review text: The problem of paired-comparison scaling when underlying scale values may change from trail to trail is studied. The Elo rating system used throughout the world for rating chess ability is presented as an analogy. The rational procedures to provide sequentially revised estimates of the reward function is proposed. The reward system is defined based on the correspondent values of the difference between current scale estimates called ratings and binary choice data. The formal definitions of the sequential paired-comparison system as well as correspondent concepts of monotone paired-comparison system, uniform system and dynamic monotone paired-comparison system are provided. The sequential paired-comparison (estimation) scheme is described to handle the case where underlying ratings may not be stationary. The set of axioms (Zero Sum, Fair Game, Constant Value and Constant Variance) is given for reward system that relate with each other reward, loss and cumulative distribution functions. Certain observations about reward systems satisfied the axioms for sequential paired-comparison are described. Asymptotic properties of the reward system with a certain condition on a distribution function are discussed. The authors propose the estimation theory for sequential paired-comparison with particular examples for two objects, one object and many objects estimation. It is shown that the Elo chess rating system was based on the special case of the presented theory (so called uniform model). The interpretation of the Bush-Mosteller linear operator learning model as two-object estimation problem with a uniform reward system is described. The results obtained can be used to determine the approximate behaviour of reward systems under conditions of fixed rating. The results of linear and diffusion approximation methods and Monte Carlo Simulation are provided for the particular examples. One-object estimation approach is proved to be formally equivalent to some methods of the item res…
Mathematics Subject Classification ID: 62J15
Mathematics Subject Classification ID: 62L12
Mathematics Subject Classification ID: 62P15
Mathematics Subject Classification ID: 65C05
zbMATH DE Number: 64527
MaRDI profile type: Publication
full work available at URL: 0022-2496(92)90036-7
OpenAlex ID: W1988792534
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description: scientific article; zbMATH DE number 64527