Majority Rule: the third author

Ben Wright considered three methods of concatenating people (Teams, Packs and Chains, RMT 9:2 432-433). Mallows (1991) proposed another concatenation: majority vote. Wright discovered that adding a low ability person to a Team (required to make unanimous decisions) lowers the success rate of the Team. Mallows discovered that, with majority decisions, even adding a competent person lowers the success rate of a competent group!

If two people of similar abilities Bn and Bm respond to an item of difficulty D independently, but are required to agree on their answer, then the combined Team ability, T, is

T = (Bn - Di) + (Bm-Di) = 2(B-D)
where B is the average of Bn and Bm.

Add a third person of similar ability and permit majority decision making. Then the Majority ability, M, under Rasch model conditions is

Measurement equation

exp(M) = = (exp(3(B-D)) + 3exp(2(B-D)))/(1 + 3exp(B-D))

1.5(B-D) <= M < 2(B-D)=T when B>=D

which accords with Mallows' pessimism.

In contrast, however, adding a third incompetent person to two other incompetents and allowing a majority vote increases their probability of success!

1.5(B-D) > M > 2(B-D)=T when B<D

Is this why democracy succeeds?

Mallows C (1991) Should you sign up that third author? Chance p. 7.

Majority rule. Wright BD, Mallows C. … Rasch Measurement Transactions, 1995, 9:3 p.443

Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

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