### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 81-93 |

Number of pages | 13 |

Journal | Combinatorics Probability and Computing |

Volume | 1 |

Issue number | 1 |

DOIs | |

State | Published - 1992 |

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### ASJC Scopus subject areas

- Applied Mathematics
- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics

### Cite this

*Combinatorics Probability and Computing*,

*1*(1), 81-93. https://doi.org/10.1017/S0963548300000080

**Three Thresholds for a Liar.** / Spencer, Joel; Winkler, Peter.

Research output: Contribution to journal › Article

*Combinatorics Probability and Computing*, vol. 1, no. 1, pp. 81-93. https://doi.org/10.1017/S0963548300000080

}

TY - JOUR

T1 - Three Thresholds for a Liar

AU - Spencer, Joel

AU - Winkler, Peter

PY - 1992

Y1 - 1992

N2 - 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.

AB - 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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U2 - 10.1017/S0963548300000080

DO - 10.1017/S0963548300000080

M3 - Article

VL - 1

SP - 81

EP - 93

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

SN - 0963-5483

IS - 1

ER -