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Three Thresholds for a Liar

  • Joel Spencer (a1) and Peter Winkler (a2)

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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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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