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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Cicalese, Ferdinando 2014. Perfect Strategies for the Ulam-Rényi Game with Multi-interval Questions. Theory of Computing Systems, Vol. 54, Issue. 4, p. 578.

    Nedev, Z. and Muthukrishnan, S. 2008. The Magnus–Derek game. Theoretical Computer Science, Vol. 393, Issue. 1-3, p. 124.

    Bohman, Tom Frieze, Alan and Wormald, Nicholas C. 2004. Avoidance of a giant component in half the edge set of a random graph. Random Structures and Algorithms, Vol. 25, Issue. 4, p. 432.

    Cicalese, Ferdinando Mundici, Daniele and Vaccaro, Ugo 2002. Least adaptive optimal search with unreliable tests. Theoretical Computer Science, Vol. 270, Issue. 1-2, p. 877.

    Marra, V. and Mundici, D. 2002. Proceedings 32nd IEEE International Symposium on Multiple- Valued Logic. p. 104.

    Pedrotti, Alberto 2002. Playing by searching: two strategies against a linearly bounded liar. Theoretical Computer Science, Vol. 282, Issue. 2, p. 285.

    Pelc, Andrzej 2002. Searching games with errors—fifty years of coping with liars. Theoretical Computer Science, Vol. 270, Issue. 1-2, p. 71.

    Aigner, Martin 1997. Finding the maximum and minimum. Discrete Applied Mathematics, Vol. 74, Issue. 1, p. 1.

    Lawler, Eugene L. and Sarkissian, Sergei 1995. An algorithm for “Ulam's Game” and its application to error correcting codes. Information Processing Letters, Vol. 56, Issue. 2, p. 89.

  • Combinatorics, Probability and Computing, Volume 1, Issue 1
  • March 1992, pp. 81-93

Three Thresholds for a Liar

  • Joel Spencer (a1) and Peter Winkler (a2)
  • DOI:
  • Published online: 01 September 2008

Motivated by the problem of making correct computations from partly false information, we study a corruption of the classic game “Twenty Questions” in which the player who answers the yes-or-no questions is permitted to lie up to a fixed fraction r of the time. The other player is allowed q arbitrary questions with which to try to determine, with certainty, which of n objects his opponent has in mind; he “wins” if he can always do so, and “wins quickly” if he can do so using only O(log n) questions.

It turns out that there is a threshold value for r below which the querier can win quickly, and above which he cannot win at all. However, the threshold value varies according to the precise rules of the game. Our “three thresholds theorem” says that when the answerer is forbidden at any point to have answered more than a fraction r of the questions incorrectly, then the threshold value is r = ½; when the requirement is merely that the total number of lies cannot exceed rq, the threshold is ⅓; and finally if the answerer gets to see all the questions before answering, the threshold drops to ¼.

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[6] Claire Kenyon and Andrew C. Yao . On evaluation of boolean functions with unreliable tests. International Journal of Foundations of Computer Science, 1(1): 110, 1990.

[7] Andrzej Pelc . Solution of ulam's problem on searching with a lie. Journal of Combinatorial Theory, Ser. A, 44: 129140, 1987.

[8] Andrzej Pelc . Searching with known error probability. Theoretical Computer Science, 63: 185202, 1989.

[9] R. L. Rivest , A. R. Meyer , D. J. Kleitman , K. Winklmann , and J. Spencer . Coping with errors in binary search procedures. Journal of Computer and System Sciences, 20: 396404, 1980.

[10] Joel Spencer . Guess a number - with lying. Mathematics Magazine, 57(2): 105108, 1984.

[11] Joel Spencer . Balancing vectors in the max norm. Combinatorica, 6: 5565, 1986.

[13] J. H. van Lint . Introduction to Coding Theory. Springer-Verlag, 1982.

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