A Computational Study of Rule Learning in “Do-It-Yourself Lottery” with Aggregate Information

  • Takashi Yamada
  • Takao Terano
Conference paper
Part of the Agent-Based Social Systems book series (ABSS, volume 11)


This chapter computationally studies Barrow’s “do-it-yourself lottery” where players choose a positive integer that is expected to be the smallest one that is not chosen by anyone else. Here, we employ and modify the rule learning framework by Stahl (Games Econ Behav 32:105–138, 2000) based on the experimental findings by Östling et al. (Am Econ J Microecon 3:1–33, 2011), and incorporate them into our simulation model to see individual and collective behavior by changing the numbers of players and the upper limit. Our main conclusion is threefold: First, the game dynamics depends on both the number of players and the upper limit. Second, a lottery with a large sensitivity parameter divides the players into winner(s) and losers. Third, finding the “stick” rule immediately makes a player a winner and imitating behavior is not observed in four-player games.


Agent-based computational economics Learning Multi-player and multi-strategy game 



This work has been revised and extended from the earlier version presented in WCSS 2012. The questions and comments are all incorporated into this study. Financial support from the Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Young Scientists (B) (24710163), from the Canon Foundation in Europe under a 2013 Research Fellowship Program (Yamada), and from JSPS and ANR under the Joint Research Project, Japan—France CHORUS Program, “Behavioral and cognitive foundations for agent-based models (BECOA)” (Terano) is gratefully acknowledged. We thank the two anonymous referees for their helpful comments and suggestions. All remaining errors are our own.


  1. 1.
    Arthur WB (1991) Designing economic agents that act like human agents: a behavioral approach to bounded rationality. Am Econ Rev 81:353–359Google Scholar
  2. 2.
    Barrow JD (2008) 100 essential things you didn’t know you didn’t know. Bodley Head, LondonGoogle Scholar
  3. 3.
    Binmore K (1998) The complexity of cooperation: agent-based models of competition and collaboration (review). J Artif Soc Soc Simul 1.
  4. 4.
    Bottazzi G, Devetag G, Dosi G (2002) Adaptive learning and emergent coordination in minority games. Simul Model Pract Theory 10:321–347CrossRefGoogle Scholar
  5. 5.
    Brenner T (2006) Agent learning representation: advice on modelling economic learning. In: Tesfatsion L, Judd KL (eds) Handbook of computational economics: agent-based computational economics, North-Holland, vol 2, pp 895–947Google Scholar
  6. 6.
    Broom M, Cannings C, Vickers GT (1997) Multi-player matrix games. Bull Math Biol 59: 931–952CrossRefGoogle Scholar
  7. 7.
    Camerer CF (2003) Behavioral game theory: experiments in strategic interaction. Princeton University Press, PrincetonGoogle Scholar
  8. 8.
    Challet D, Zhang Y-C (1997) Emergence of cooperation and organization in an evolutionary game. Phys A 246:407–418CrossRefGoogle Scholar
  9. 9.
    Challet D, Zhang Y-C (1998) On the minority game: analytical and numerical studies. Phys A 256:514–532CrossRefGoogle Scholar
  10. 10.
    Crawford VP, Costa-Gomes MA, Iriberri N (2013) Structural models of nonequilibrium strategic thinking: theory, evidence, and application. J Econ Lit 51:5–62CrossRefGoogle Scholar
  11. 11.
    Duffy J (2006) Agent-based models and human subject experiments. In: Tesfatsion L, Judd KL (eds) Handbook of computational economics: agent-based computational economics, North-Holland, vol 2, pp 949–1012Google Scholar
  12. 12.
    Duffy J, Nagel R (1997) On the robustness of behavior in experimental “beauty contest” games. Econ J 107:1684–1700CrossRefGoogle Scholar
  13. 13.
    Eichberger J, Vinogradov D (2008) Least unmatched price auctions: a first approach. Department of Economics, University of Heidelberg, Discussion Paper Series No. 471Google Scholar
  14. 14.
    Erev I, Roth AE (2007) Multi-agent learning and the descriptive value of simple models. Artif Intell 171:423–428CrossRefGoogle Scholar
  15. 15.
    Gallice A (2009) Lowest unique bid auctions with signals. Collegio Carlo Alberto Working Paper 112Google Scholar
  16. 16.
    Gintis H (2009) Bounds of reason. Princeton University Press, PrincetonGoogle Scholar
  17. 17.
    Goeree JK, Yariv L (2010) An experimental study of collective deliberation. Econometrica 79:893–921Google Scholar
  18. 18.
    Halpern JY (2010) Computer science and game theory. In: Durlauf SN, Blume LE (eds) Game theory. Palgrave Macmillan, London, pp 48–65Google Scholar
  19. 19.
    Hare M, Deadman P (2004) Further towards a taxonomy of agent-based simulation models in environmental management. Math Comput Simul 64:25–40CrossRefGoogle Scholar
  20. 20.
    Haruvy E, Stahl DO (2007) Equilibrium selection and bounded rationality in symmetric normal form games. J Econ Behav Org 62:98–119CrossRefGoogle Scholar
  21. 21.
    Ho T-H, Camerer CF, Weigelt K (1998) Iterated dominance and iterated best response in experimental “p-beauty contests”. Am Econ Rev 88:947–969Google Scholar
  22. 22.
    Houba H, van der Laan D, Veldhuizen D (2011) Endogenous entry in lowest-unique sealed-bid auctions. Theory Decis 71:269–295CrossRefGoogle Scholar
  23. 23.
    Judd KL (2006) Computationally intensive analyses in economics. In: Tesfatsion L, Judd KL (eds) Handbook of computational economics: agent-based computational economics, vol 2, pp 881–893Google Scholar
  24. 24.
    Klügl F, Bazzan ALC (2012) Agent-based modeling and simulation. AI Magazine Fall:29–40Google Scholar
  25. 25.
    McCabe KA, Mukherji A, Runkle DE (2000) An experimental study of information and mixed-strategy play in the three-person matching-pennies game. Econ Theory 15:421–462CrossRefGoogle Scholar
  26. 26.
    Matsumura M, Ikegami T (1998) Evolution of strategies in the three-person iterated prisoner’s dilemma game. J Theory Biol 195:53–67CrossRefGoogle Scholar
  27. 27.
    Nagel R (1995) Unraveling in guessing games: an experimental study. Am Econ Rev 85: 1313–1326Google Scholar
  28. 28.
    Nagel R (2008) Experimental beauty contest games: levels of reasoning and convergence to equilibrium. In: Plott CR, Smith VL (eds) Handbook of experimental economics results, North-Holland, vol 1, pp 391–410CrossRefGoogle Scholar
  29. 29.
    Östling R, Wang JT, Chou EY, Camerer CF (2011) Testing game theory in the field: Swedish LUPI lottery games. Am Econ J Microecon 3:1–33CrossRefGoogle Scholar
  30. 30.
    Pigolotti S, Bernhardsson S, Juul J, Galster G, Vivo P (2012) Equilibrium strategy and population-size effects in lowest unique bid auctions. Phys Rev Lett 108:088701-1–088701-5Google Scholar
  31. 31.
    Platkowski T (2004) Evolution of population playing mixed multiplayer games. Math Comput Model 39:981–989CrossRefGoogle Scholar
  32. 32.
    Rapoport A, Otsubo H, Kim B, Stein WE (2009) Unique bid auction games. Jena Economic Research Paper 2009-005Google Scholar
  33. 33.
    Raviv Y, Virag G (2009) Gambling by auction. Int J Ind Org 27:369–378CrossRefGoogle Scholar
  34. 34.
    Sandholm T (2007) Perspectives on multiagent learning. Artif Intell 171:382–391CrossRefGoogle Scholar
  35. 35.
    Shoham Y, Powers R, Grenager T (2007) If multi-agent learning is the answer, what is the question? Artif Intell 171:365–377CrossRefGoogle Scholar
  36. 36.
    Stahl DO (1999) Evidence based rules and learning in symmetric normal-form games. Int J Game Theory 28:111–130CrossRefGoogle Scholar
  37. 37.
    Stahl DO (2000) Rule learning in symmetric normal-form games: theory and evidence. Games Econ Behav 32:105–138CrossRefGoogle Scholar
  38. 38.
    Szilagyi MN (2003) An investigation of N-person prisoners’ dilemmas. Complex Syst 14: 155–174Google Scholar
  39. 39.
    Vu T, Powers R, Shoham Y (2006) Learning in games with more than two players. In: Fifth international joint conference on autonomous agents and multi agent systems (AAMAS 2006)Google Scholar
  40. 40.
    Weizsäcker G (2010) Do we follow others when we should? A simple test of rational expectations. Am Econ Rev 100:2340–2360CrossRefGoogle Scholar
  41. 41.
    Wilkinson N, Klaes M (2012) An introduction to behavioral economics, 2nd edn. Palgrave Macmillan, LondonGoogle Scholar
  42. 42.
    Winkler P (2004) Mathematical puzzles: a connoisseur’s collection. A K Peters, WellesleyGoogle Scholar
  43. 43.
    Zhao J, Szilagyi MN, Szidarovszky F (2008) An n-person battle of sexes game. Phys A 387:3669–3677CrossRefGoogle Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.Department of Computational Intelligence and Systems Science, Interdisciplinary Graduate School of Science and EngineeringTokyo Institute of TechnologyYokohamaJapan

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