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

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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