Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-25T18:42:48.242Z Has data issue: false hasContentIssue false
  • English
  • Français

Three Thresholds for a Liar

Published online by Cambridge University Press:  12 September 2008

Joel Spencer  and
Peter Winkler
Joel Spencer
Affiliation:
NYU-Courant Institute, 251 Mercer St., New York, NY 10012
Peter Winkler
Affiliation:
Bellcore, 445 South St., Morristown, NJ 07962-1910
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 1 , Issue 1 , March 1992 , pp. 81 - 93
Copyright
Copyright © Cambridge University Press 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Aslam, Javed and Dhagat, Aditi. Searching in the presence of linearly bounded errors. In Proceedings of the 23rd Annual ACM Symposium of Theory of Computing, 1991.CrossRefGoogle Scholar
[2] Berlekamp, E.. Block Coding with Noiseless Feedback. PhD thesis, M.I.T., 1964.Google Scholar
[3] Feige, U., Peleg, D., Raghavan, P., and Upfal, E.. Computing with unreliable information. In Symposium on Theory of Computing, pages 128–137, 1990.Google Scholar
[4] Frazier, Michael. Searching with a non-constant number of lies. 1990.Google Scholar
[5] Hall, Marshall. Combinatorial Theory (2nd ed.). Wiley, 1986.Google Scholar
[6] Kenyon, Claire and Yao, Andrew C.. On evaluation of boolean functions with unreliable tests. International Journal of Foundations of Computer Science, 1(1): 1–10, 1990.CrossRefGoogle Scholar
[7] Pelc, Andrzej. Solution of ulam's problem on searching with a lie. Journal of Combinatorial Theory, Ser. A, 44: 129–140, 1987.CrossRefGoogle Scholar
[8] Pelc, Andrzej. Searching with known error probability. Theoretical Computer Science, 63: 185–202, 1989.CrossRefGoogle Scholar
[9] Rivest, R. L., Meyer, A. R., Kleitman, D. J., Winklmann, K., and Spencer, J.. Coping with errors in binary search procedures. Journal of Computer and System Sciences, 20: 396–404, 1980.CrossRefGoogle Scholar
[10] Spencer, Joel. Guess a number - with lying. Mathematics Magazine, 57(2): 105–108, 1984.Google Scholar
[11] Spencer, Joel. Balancing vectors in the max norm. Combinatorica, 6: 55–65, 1986.CrossRefGoogle Scholar
[12] Ulam, S. M.. Adventures of a Mathematician. Charles Scribner's Sons, 1976.Google Scholar
[13] Lint, J. H. van. Introduction to Coding Theory. Springer-Verlag, 1982.CrossRefGoogle Scholar