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