Q-Ary Ulam-Renyi Game with Constrained Lies

Abstract

The Ulam-Rényi game is a classical model for the problem of finding an unknown number in a finite set using as few question as possible when up to a finite number e of the answers may be lies. In the variant, we consider in this paper, questions with q many possible answers are allowed, q fixed and known beforehand, and lies are constrained as follows: Let ${\mathcal Q} = \{0,1,\dots, q-1\}$ be the set of possible answers to a q-ary question. For each $k \in {\mathcal Q}$ when the sincere answer to the question is k, the responder can choose a mendacious answer only from a set $L(k) \subseteq {\mathcal Q} \setminus \{k\}.$ For each $k \in {\mathcal Q} ,$ the set L(k) is fixed before the game starts and known to the questioner. The classical q-ary Ulam-Rényi game, in which the responder is completely free in choosing the lies, in our setting corresponds to the particular case $L(k) = {\mathcal Q} \setminus \{k\},$ for each $k \in {\mathcal Q}.$ The problem we consider here, is suggested by the counterpart of the Ulam-Rényi game in the theory of error-correcting codes, where (the counterparts of) lies are due to the noise in the channel carrying the answers. We shall use our assumptions on noise and its effects (as represented by the constraints L(k) over the possible error patterns) with the aim of producing the most efficient search strategies. We solve the problem by assuming some symmetry on the sets L(k): specifically, we assume that there exists a constant d ≤q–1 such that |L(k)| = d for each k, and the number of indices j such that k ∈L(j) is equal to d. We provide a lower bound on the number of questions needed to solve the problem and prove that in infinitely many cases this bound is attained by (optimal) search strategies. Moreover we prove that, in the remaining cases, at most one question more than the lower bound is always sufficient to successfully find the unknown number. Our results are constructive and search strategies are actually provided. All our strategies also enjoy the property that, among all the possible adaptive strategies, they use the minimum amount of adaptiveness during the search process.