A plurality problem with three colors and query size three
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The Plurality problem - introduced by Aigner A2004 - has many variants. In this article we deal with the following version: suppose we are given n balls, each of them colored by one of three colors. A plurality ball is one such that its color class is strictly larger than any other color class. Questioner wants to find a plurality ball as soon as possible or state there is no, by asking triplets (or k-sets, in general), while Adversary partition the triplets into color classes as an answer for the queries and wants to postpone the possibility of determining a plurality ball (or stating there is no). We denote by A_p(n,3) the largest number of queries needed to ask if both play optimally (and Questioner asks triplets). We provide an almost precise result in case of even n by proving that for n > 4 even we have 3/4n-2 < A_p(n,3) <3/4n-1/2, and for n > 3 odd we have 3/4n-O( n) < A_p(n,3) <3/4n-1/2. We also prove some bounds on the number of queries needed to ask for larger k.
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