The bridge tournament problem and calibration designs for comparing pairs of objects

Abstract

The c lassical tournam e nt problem caiJ s for arranging v individuals int o tea ms o f /) p layers so that a pla yer is teamed th e sa m e number of times with each of th e other pla ye rs and a lso that eac h player is pitted eq ually often aga in st each of th e other players. The play of th e tuurn a m en t resu lt s in th e determination of difference in p e rformance of th e various pairings of th e g roups. In th e s pecial case when p = 2 each team co n s ist s of two players and th e d es igri s are call e d brid ge tourn a me nt designs .In high prec is iun ca libration one ca n m eas ure only the difference be tw ee n t wo nominally eq ual g ro up s S(l that if u objects a re to be int e rco mpare d in groups of fJ objects, th en th e so lution s tu th e to urnam e nt probl e m provid e sc hedul es for th e gl'll uping. Th ese des ig ns are useful in we ig hing a nd any o th e r m eas ul'e m c nt s where th e obj ec ts tu b e meas ure d ca n b e co mbin ed intu gl'll ups without loss of prec is ion or acc uracy ill th e compari sons.Tili s pape r pre se nt s ge ne ral m e thud s fill' co nstru c tin g of brid ge lourn a m e nt d es ign s, i. e ., for Lh t' case wh c n p = 2, fur all /1 ~ 50.Key Wurd s : Ca libration , ca li bration designs. co mbin a tori a l analysis , diffe re nce se ts, e xpe riment d es ig ns' , in c omplete bloc k d es igns, tournam e nt s , weighing de s ign s.

A survey of partial difference sets

“…nonabelian). For a subset S of a group G, let S it) = {g~ : g E S} for any integer t. PDSs were named by Chakravarti [8] but they were introduced by Bose and Cameron [9] in their studies of calibration designs and the bridge tournament problem. Although a systematic study of PDSs was started by Ma [10], [11] as a generalization of difference sets, there were a lot of earlier results written in terms of strongly regular graphs or related topics, e.g., Delsarte [12]- [14], Camion [15], Bridges and Mena [16]- [17] and Calderbank and Kantor [18].…”

Partitionable sets, almost partitionable sets, and their applications

“…Bose and Cameron [11] in 1965 and Baker [8] in 1975 independently established the existence of ZCPS‐Wh for any prime . Zhang and Chang [41] in 2009 presented the same result for PS.…”

Designs for the calibration of standards of mass

“…A direct check on the limiting mean of the measurement (2) and use the method of Lagrangian multipliers (with multipliers 2A.) to minimize the function *In all cases treated here a single restraint is sufficient.…”