Was the Rasch model Almost the Zermelo model?

Here is Rasch's version of the familiar Rasch model: (Rasch, 1980, p. 168):

Here is Zermelo's (1929) model:

Both are multiplicative forms of the Rasch model. What is the difference? Rasch is contrasting persons with items. Zermelo is modeling paired comparisons. u_r and u_s are the strengths of two chess players. Zermelo notes that this model converts the strengths of the chess-players into probabilities, u_rs. These strengths are estimated from their tournaments scores. A draw is accommodated by adding 0.5 to the player's raw score. Zermelo omits to mention how he would use his model to predict draws. In current practice, the prediction of draws is facilitated by using a rating-scale model (RMT 11:3, p. 584). Zermelo, however, perceives that his model is robust against unplayed games (missing data) and able to accommodate linked tournaments! (Probably the first time these attributes of a Rasch model are noticed.)

Why has Zermelo's model been ignored? His estimation technique is arduous, but worse, his explanation of the inferential meaning of the parameters is deficient. He fails to emphasize that u_r /u_s are the odds of success of player r relative to player s. Further, his paper lacks the fundamental insight that log_e(u_r) linearizes "Player strength", permitting the production of a useful picture. The Figure shows such a picture, plotted from Zermelo's own example analysis, which, we can see, lacks the drama which was present at the Tournament (Soltis, 1975). Zermelo's results are the same as those obtained using Facets with a dichotomous model and half-weighting of draws. A modern analyst, however, is more likely to use a rating scale model, so that draws can be predicted explicitly. For comparison, the player measures for a paired rating-scale model are shown in the Figure and are seen to differ little from Zermelo's estimates.

When Bradley and Terry (1952) independently formulated Zermelo's model, they observed that: "If the estimates are converted to logarithms, the values log_e [u_r] occur on a linear scale and permit overall comparisons of the experimental treatments [or chess players]. Any considerations of differences among treatments [or players] should be based on the values of the log [u_r]'s." (p. 326)

Bradley, R.A., and Terry, M.E. Rank analysis of incomplete block designs. I. The method of paired comparisons, Biometrika. 1952, 39, 324-45.

Soltis, A. (1975) The Great Chess Tournaments and Their Stories. Radnor, Pa.: Chilton.

Zermelo, E. (1929) The calculation of tournament results as a maximum-likelihood problem [German]. Mathematische Zeitschrift, 29, 436-460.

Was the Rasch model Almost the Zermelo model? Zermelo, E. … Rasch Measurement Transactions, 2000, 14:2 p. 754.

Rasch-Related Resources: Rasch Measurement YouTube Channel
Rasch Measurement Transactions & Rasch Measurement research papers - free	An Introduction to the Rasch Model with Examples in R (eRm, etc.), Debelak, Strobl, Zeigenfuse	Rasch Measurement Theory Analysis in R, Wind, Hua	Applying the Rasch Model in Social Sciences Using R, Lamprianou	El modelo métrico de Rasch: Fundamentación, implementación e interpretación de la medida en ciencias sociales (Spanish Edition), Manuel González-Montesinos M.
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar	Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch	Rasch Models for Measurement, David Andrich	Constructing Measures, Mark Wilson	Best Test Design - free, Wright & Stone Rating Scale Analysis - free, Wright & Masters
Virtual Standard Setting: Setting Cut Scores, Charalambos Kollias	Diseño de Mejores Pruebas - free, Spanish Best Test Design	A Course in Rasch Measurement Theory, Andrich, Marais	Rasch Models in Health, Christensen, Kreiner, Mesba	Multivariate and Mixture Distribution Rasch Models, von Davier, Carstensen
Rasch Books and Publications: Winsteps and Facets
Applying the Rasch Model (Winsteps, Facets) 4th Ed., Bond, Yan, Heene	Advances in Rasch Analyses in the Human Sciences (Winsteps, Facets) 1st Ed., Boone, Staver	Advances in Applications of Rasch Measurement in Science Education, X. Liu & W. J. Boone	Rasch Analysis in the Human Sciences (Winsteps) Boone, Staver, Yale	Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes	Rasch Models for Solving Measurement Problems (Facets), George Engelhard, Jr. & Jue Wang	Statistical Analyses for Language Testers (Facets), Rita Green	Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments (Facets), George Engelhard, Jr. & Stefanie Wind	Aplicação do Modelo de Rasch (Português), de Bond, Trevor G., Fox, Christine M
Exploring Rating Scale Functioning for Survey Research (R, Facets), Stefanie Wind	Rasch Measurement: Applications, Khine	Winsteps Tutorials - free Facets Tutorials - free	Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre	Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan

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