Two-batch liar games on a general bounded channel

TL;DR

This paper extends the liar game to channels with bounded lie strings, analyzing two-batch strategies and providing asymptotic bounds for the maximum search space size, generalizing prior results and addressing pathological variants.

Contribution

It introduces a two-batch liar game model with bounded lie channels and derives asymptotic bounds for winning strategies, extending previous work and unifying solutions for pathological variants.

Findings

Maximum search space size scales as ~ t^{q+k}/(E_k(C) * binom(q,k))

Provides asymptotically perfect two-batch codes for the channel C

Generalizes prior liar game results to bounded lie channels

Abstract

We consider an extension of the 2-person R\'enyi-Ulam liar game in which lies are governed by a channel $C$, a set of allowable lie strings of maximum length $k$. Carole selects $x\in[n]$, and Paul makes $t$-ary queries to uniquely determine $x$. In each of $q$ rounds, Paul weakly partitions $[n]=A_0\cup >... \cup A_{t-1}$ and asks for $a$ such that $x\in A_a$. Carole responds with some $b$, and if $a\neq b$, then $x$ accumulates a lie $(a,b)$. Carole's string of lies for $x$ must be in the channel $C$. Paul wins if he determines $x$ within $q$ rounds. We further restrict Paul to ask his questions in two off-line batches. We show that for a range of sizes of the second batch, the maximum size of the search space $[n]$ for which Paul can guarantee finding the distinguished element is $\sim t^{q+k}/(E_k(C)\binom{q}{k})$ as $q\to\infty$, where $E_k(C)$ is the number of lie strings in $C$ of maximum length $k$. This generalizes previous work of Dumitriu and Spencer, and of Ahlswede, Cicalese, and Deppe. We extend Paul's strategy to solve also the pathological liar variant, in a unified manner which gives the existence of asymptotically perfect two-batch adaptive codes for the channel $C$.