On Ranking the Players in a 3-Player Tournament

摘要

M. Sobel and G. Weiss have proposed sequential selection (identification and ranking) procedures for 3-player tournaments. In their formulation of the problem they permit only one of the three terminal decisions: 'Player i is 'best'.' In the formulation described in the present paper, a fourth decision is also permitted, namely, 'There is no clear-cut 'best' player in the tournament'--to cope with those situations for which the possibility exists that (say) Player 1 is better than Player 2, Player 2 is better than Player 3, and Player 3 is better than Player 1. For this 4-decision formulation, a single-stage ranking procedure is proposed which guarantees that certain requirements on the probability of a correct selection will be achieved (no matter which one of the four States of Nature is the true one); for this same formulation, a sequential identification procedure is proposed which guarantees that certain more restrictive requirements on the probability of a correct selection will be achieved. The sequential identification procedure can be regarded as a generalization of one of the sequential procedures proposed by Sobel and Weiss. (Author)