We consider a natural generalization of the classical ruin problem to more than two parties. Our “ruin” problem, which we will call the (k, I)-game, starts with k players each having I units as its initial capital. At each round of the game, all remaining k′ players pay 1/k′th unit as game fee, play the game, and one of the players wins and receives the combined game fees of 1 unit. A player who cannot pay the next game fee goes bankrupt, and the game terminates when all players but one are bankrupt. We analyze the length of the game, that is, the number of rounds executed until the game terminates, and give upper and lower bounds for the expected game length.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. AsmOO.
    S. Asmussen, Ruin Probabilities, World Scientific, 2000.Google Scholar
  2. DGW98.
    Carlos Domingo, Ricard Gavaldà, and Osamu Watanabe, Practical algorithms for on-line sampling, in Proc. of the First International Conference on Discovery Science, Lecture Notes in Artificial Intelligence 1532, 150–162 (1998).Google Scholar
  3. DWY98.
    C. Domingo, O. Watanabe, T. Yamazaki, A role of constraint in self-organization, in Proc. Workshop on Randomization and Approximation Techniques in Computer Science (RANDOM98), Lecture Notes in Computer Science 1518, Springer-Verlag, 307–318 (1998).CrossRefGoogle Scholar
  4. Fel57.
    W. Feller, An Introduction to Probability Theory and Its Applications, Wiley, New york, 1957.zbMATHGoogle Scholar
  5. It73.
    Y. Itoh, On a ruin problem with interaction, Ann. Inst. Statist. Math. 25, 635–641 (1973).zbMATHCrossRefMathSciNetGoogle Scholar
  6. IM98.
    Y. Itoh and H. Maehara, A variation to the ruin problem, Math. Japonica 47(1), 97–102 (1998).zbMATHMathSciNetGoogle Scholar
  7. vdM73.
    C. von der Malsburg, Self-organization of orientation sensitive cells in the striate cortex, Kybernetik 14, 85–100 (1973).CrossRefGoogle Scholar
  8. WY97.
    O. Watanabe and T. Yamazaki, Orientation selectivity: An approach from theoretical computer science, TR97-0008, Comp. Sci. Dept., Tokyo Institute of Technology, Tokyo, November 1997, Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Kazuyuki Amano
    • 1
  • John Tromp
    • 2
  • Paul M. B. Vitányi
    • 3
  • Osamu Watanabe
    • 4
  1. 1.GSISTohoku UniversityJapan
  2. 2.CWIThe Netherlands
  3. 3.CWI and University of AmsterdamThe Netherlands
  4. 4.Dept. of Math. & Comp. Sci.Tokyo Inst. of Tech.Japan

Personalised recommendations