Intransitiveness in Games and Random Walks

  • Alberto Baldi
  • Giulia Cencetti
  • Duccio Fanelli
  • Franco BagnoliEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11938)


In this paper we introduce the concept of intransitiveness for games, which is the condition for which there is no first-player winning strategy, and the second player can statistically win. We show that the game can be converted into a random walk on a graph, i.e., a Markov process, and therefore we can extend the intransitiveness concept to such systems.

The end of the game generally consists in the appearance of a pattern chosen by one of the player. In the language of random walk this corresponds to an absorbing trap, since once that the game has reached this condition the game itself comes to an end. Therefore, the intransitiveness of the game can be mapped into a problem of competition among traps.

We analyse in details this problem for the Penney game (an extension of the heads or tails game which is intransitive for sequences longer than three), for walks on a circle and for a scale-free network, reminiscent of the structure of the world wide web. We also introduce several variations: traps can be partially absorbing, the walk can be biased and the initial distribution can be arbitrary.

We found that the intransitiveness concept can be quite useful for characterizing the properties of a graph, and that the consequences of the above mentioned extensions are not trivial.


  1. 1.
    Albert, R., Barabási, A.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74(1), 47 (2002)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Barabási, A.-L., Albert, R., Jeong, H.: Scale-free characteristics of random networks: the topology of the world-wide web. Phys. A Stat. Mech. Its Appl. 281(1–4), 69–77 (2000)CrossRefGoogle Scholar
  3. 3.
    Barabási, A.-L., Bonabeau, E.: Scale-free networks. Sci. Am. 288(5), 60–69 (2003)CrossRefGoogle Scholar
  4. 4.
    Brin, S., Page, L.: The anatomy of a large-scale hypertextual web search engine. In: Seventh International World-Wide Web Conference (WWW), Brisbane, Australia, 14–18 April 1998 (1998)Google Scholar
  5. 5.
    Cencetti, G., Bagnoli, F., Di Patti, F., Fanelli, D.: The second will be first: competition on directed networks. Sci. Rep. 6, 27116 (2016)CrossRefGoogle Scholar
  6. 6.
    Langville, A.N., Meyer, C.D.: Google’s PageRank and Beyond: The Science of Search Engine Rankings. Princeton University Press, Princeton (2011)zbMATHGoogle Scholar
  7. 7.
    Penney, W.: Problem 95: penney-ante. J. Recreat. Math. 2, 241 (1969)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Alberto Baldi
    • 1
  • Giulia Cencetti
    • 2
  • Duccio Fanelli
    • 1
    • 3
  • Franco Bagnoli
    • 1
    • 3
    Email author
  1. 1.Department of Physics and Astronomy and CSDCUniversity of FlorenceSesto FiorentinoItaly
  2. 2.Mobs Lab, Fondazione Bruno KesslerPovoItaly
  3. 3.INFN, sez. FirenzeFlorenceItaly

Personalised recommendations