A plurality problem with three colors and query size three

TL;DR

This paper investigates the optimal number of queries needed to find a plurality ball among three colors using triplet questions, providing tight bounds for even and odd numbers of balls.

Contribution

It establishes nearly exact bounds for the number of queries required in the three-color plurality problem with triplet queries, extending to larger query sizes.

Findings

For even n ≥ 4, A_p(n,3) is approximately 3/4 n.

For odd n ≥ 3, A_p(n,3) is approximately 3/4 n with a logarithmic correction.

Provides bounds for larger k in the plurality problem.

Abstract

The Plurality problem - introduced by Aigner \cite{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 \textit{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 $n \ge 4$ even we have $$\frac{3}{4}n-2 \le A_p(n,3) \le \frac{3}{4}n-\frac{1}{2},$$ and for $n \ge 3$ odd we have $$\frac{3}{4}n-O(\log n) \le A_p(n,3) \le \frac{3}{4}n-\frac{1}{2}.$$ We also prove some bounds on the number of queries needed to ask for larger $k$.