Two-batch liar games on a general bounded channel

Abstract

We consider an extension of the 2-person Rényi–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 ∈ [ n ] , and Paul makes t-ary queries to uniquely determine x. In each of q rounds, Paul weakly partitions [ n ] = A 0 ∪ ⋯ ∪ A t − 1 and asks for a such that x ∈ A a . Carole responds with some b, and if a ≠ 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 ∼ t q + k / ( E k ( C ) ( q k ) ) as q → ∞ , 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.