## How Much Can You Win When Your Adversary is Handicapped? (2000)

Venue: | PRESENTED AT THE SYMPOSIUM "NUMBERS, INFORMATION AND COMPLEXITY", BIELEFELD, OCTOBER 8 - 11, 1998 |

Citations: | 6 - 0 self |

### BibTeX

@INPROCEEDINGS{Staiger00howmuch,

author = {Ludwig Staiger},

title = {How Much Can You Win When Your Adversary is Handicapped?},

booktitle = {PRESENTED AT THE SYMPOSIUM "NUMBERS, INFORMATION AND COMPLEXITY", BIELEFELD, OCTOBER 8 - 11, 1998},

year = {2000},

pages = {403--412},

publisher = {Kluwer}

}

### OpenURL

### Abstract

We consider infinite games where a gambler plays a coin-tossing game against an adversary. The gambler puts stakes on heads or tails, and the adversary tosses a fair coin, but has to choose his outcome according to a previously given law known to the gambler. In other words, the adversary is not allowed to play all infinite heads-tails-sequences, but only a certain subset F of them. We present an algorithm for the player which, depending on the structure of the set F , guarantees an optimal exponent of increase of the player's capital, independently on which one of the allowed heads-tails-sequences the adversary chooses. Using the known upper bound on the exponent provided by the maximum Kolmogorov complexity of sequences in F we show the optimality of our result.

### Citations

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Citation Context ...also our main result. Most of the results presented here are proved in [St98]. For the necessary background in computability, random sequences and Kolmogorov complexity we refer the reader to [Sc71], =-=[LV93]-=- and [Ca94]. For the definition of Hausdorff dimension and their properties see e.g. [Ed90, Fa90]. 1 Notation and Definitions By IN = f0;1;2; : : :g we denote the set of natural numbers. We consider t... |

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Citation Context ...rov complexity. Definition 2 The lower Kolmogorov complexity of an infinite string x 2 f0;1g w is the value k(x) := liminf n! K U (x=n) n : Utilizing Levin's universal semicomputable semimeasure (cf. =-=[ZL70]-=- or [LV93]) it was shown in [Ry93] that the exponent l V (x) is bounded from above by 1 \Gamma k(x) provided the gambler plays according to a computable strategy. Lemma 2 (Upper bound by Kolmogorov co... |

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Citation Context ...e state also our main result. Most of the results presented here are proved in [St98]. For the necessary background in computability, random sequences and Kolmogorov complexity we refer the reader to =-=[Sc71]-=-, [LV93] and [Ca94]. For the definition of Hausdorff dimension and their properties see e.g. [Ed90, Fa90]. 1 Notation and Definitions By IN = f0;1;2; : : :g we denote the set of natural numbers. We co... |

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Citation Context ...in result. Most of the results presented here are proved in [St98]. For the necessary background in computability, random sequences and Kolmogorov complexity we refer the reader to [Sc71], [LV93] and =-=[Ca94]-=-. For the definition of Hausdorff dimension and their properties see e.g. [Ed90, Fa90]. 1 Notation and Definitions By IN = f0;1;2; : : :g we denote the set of natural numbers. We consider the space f0... |

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Citation Context ... that several papers investigated the relationship between the Kolmogorov complexity of infinite strings and size measures known from information theory and fractal geometry. It turned out in [Ry86], =-=[St93]-=- and [St98] that the Hausdorff dimension of subsets F ae f0;1g w is closely related to sup x2F k(x). Definition 3 The Hausdorff dimension of a set F ` f0;1g w , dimF , is the smallest real number as0 ... |

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Citation Context ...ntioning that several papers investigated the relationship between the Kolmogorov complexity of infinite strings and size measures known from information theory and fractal geometry. It turned out in =-=[Ry86]-=-, [St93] and [St98] that the Hausdorff dimension of subsets F ae f0;1g w is closely related to sup x2F k(x). Definition 3 The Hausdorff dimension of a set F ` f0;1g w , dimF , is the smallest real num... |

42 | A tight upper bound on Kolmogorov complexity and uniformly optimal prediction
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(Show Context)
Citation Context ...the fourth section we discuss the computability requirements which we have to put on our constraints F ` f0;1g w . Here we state also our main result. Most of the results presented here are proved in =-=[St98]-=-. For the necessary background in computability, random sequences and Kolmogorov complexity we refer the reader to [Sc71], [LV93] and [Ca94]. For the definition of Hausdorff dimension and their proper... |

16 |
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Citation Context ...ower Kolmogorov complexity of an infinite string x 2 f0;1g w is the value k(x) := liminf n! K U (x=n) n : Utilizing Levin's universal semicomputable semimeasure (cf. [ZL70] or [LV93]) it was shown in =-=[Ry93]-=- that the exponent l V (x) is bounded from above by 1 \Gamma k(x) provided the gambler plays according to a computable strategy. Lemma 2 (Upper bound by Kolmogorov complexity) Let V be a computable ca... |

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