Optimal strategies against a liar

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We consider the following scenario: There are two individuals, say Q (Questioner) and R (Responder), involved in a search game. Player R chooses a number, say x, from the set S = {1, . . . , M}. Player Q has to find out x by asking questions of type: "which one of the sets A1, A2, . . . , Aq, does x belong to?", where the sets A1, . . . , Aq constitute a partition of S. Player R answers "i" to indicate that the number x belongs to Ai. We are interested in the least number of questions player Q has to ask in order to be always able to correctly guess the number x, provided that R can lie at most e times. The case e = 0 obviously reduces to the classical q-ary search, and the necessary number of questions is [logqM]. The case q = 2 and e ≥ 1 has been widely studied, and it is generally referred to as Ulam's game. In this paper we consider the general case of arbitrary q ≥ 1. Under the assumption that player R is allowed to lie at most twice throughout the game, we determine the minimum number of questions Q needs to ask in order to successfully search for x in a set of cardinality M = qi, for any i ≥ 1. As a corollary, we obtain a counterexample to a recently proposed conjecture of Aigner, for the case of an arbitrary number of lies. We also exactly solve the problem when player R is allowed to lie a fixed but otherwise arbitrary number of times e, and M = qi, with i not too large with respect to q. For the general case of arbitrary M, we give fairly tight upper and lower bounds on the number of the necessary questions. © 2000 Elsevier Science B.V. All rights reserved.




Cicalese, F., & Vaccaro, U. (2000). Optimal strategies against a liar. Theoretical Computer Science, 230(1–2), 167–193. https://doi.org/10.1016/S0304-3975(99)00044-4

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