On the Rényi–Ulam game with restricted size queries

Abstract

We investigate the following version of the well-known Rényi–Ulam game. Two players – the Questioner and the Responder – play against each other. The Responder thinks of a number from the set $\{1,\dots ,n\}$, and the Questioner has to find this number. To do this, he can ask whether a chosen set of at most $k$ elements contains the thought number. The Responder answers with YES or NO immediately, but during the game, he may lie at most $\ell$ times. The minimum number of queries needed for the Questioner to surely find the unknown element is denoted by $R{U}_{\ell }^k\left(n\right)$. First, we develop a highly effective tool that we call Convexity Lemma. By using this lemma, we give a general lower bound of $R{U}_{\ell }^k\left(n\right)$ and an upper bound which differs from the lower one by at most $2\ell +1$. We also give its exact value when $n$ is sufficiently large compared to $k$. With these, we managed to improve and generalize the results obtained by Meng, Lin, and Yang in a 2013 paper about the case $\ell =1$.