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Super- and submodularity of stopping games with random observations

  • Svetlana BoyarchenkoEmail author
Research Article
  • 20 Downloads

Abstract

Models of learning and experimentation based on two-armed Poisson bandits addressed several important aspects related to strategic and motivational learning, but they are not suitable to study effects that accumulate over time. We propose a new class of models of strategic experimentation which are almost as tractable as exponential models, but incorporate such realistic features as dependence of the expected rate of news arrival on the time elapsed since the start of an experiment. In these models, the experiment is stopped before news is realized whenever the rate of arrival of news reaches a critical level. This leads to longer experimentation times for experiments with possible breakthroughs than for equivalent experiments with failures. We also show that the game with conclusive failures is supermodular, and the game with conclusive breakthroughs is submodular.

Keywords

Stopping games Supermodular games Time-inhomogeneous Poisson process 

JEL Classification

C73 C61 D81 

Notes

References

  1. Amir, R.: Continuous stochastic games of capital accumulation with convex transitions. Games Econ. Behav. 15, 111–131 (1996a)CrossRefGoogle Scholar
  2. Amir, R.: Cournot oligopoly and the theory of supermodular games. Games Econ. Behav. 15, 132–148 (1996b)CrossRefGoogle Scholar
  3. Amir, R., Lazzati, N.: Network effects, market structure and industry performance. J. Econ. Theory 146, 2389–2419 (2011)CrossRefGoogle Scholar
  4. Balbus, Ł., Reffett, K., Woźny, Ł.: A constructive study of markov equilibria in stochastic games with strategic complementarities. J. Econ. Theory 150, 815–840 (2014)CrossRefGoogle Scholar
  5. Bergemann, D., Välimäki, J.: Learning and strategic pricing. Econometrica 64, 1125–1149 (1996)CrossRefGoogle Scholar
  6. Bergemann, D., Välimäki, J.: Experimentation in markets. Rev. Econ. Stud. 67, 213–234 (2000)CrossRefGoogle Scholar
  7. Bergemann, D., Välimäki, J.: Dynamic price competition. J. Econ. Theory 127, 232–263 (2006)CrossRefGoogle Scholar
  8. Bergemann, D., Välimäki, J.: Bandit problems. In: Durlauf, S.N., Blume, L.E. (eds.) The New Palgrave Dictionary of Economics. Palgrave Macmillan, Basingstoke (2008)Google Scholar
  9. Bolton, P., Harris, C.: Strategic experimentation. Econometrica 67, 349–374 (1999)CrossRefGoogle Scholar
  10. Boyarchenko, S.: Strategic experimentation with humped bandits. Working paper. Available at SSRN. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3174107 (2018). Accessed 16 May 2019
  11. Boyarchenko, S.I., Levendorksiĭ, S.Z.: Preemption games under Lévy uncertainty. Games Econ. Behav. 88, 354–380 (2014)CrossRefGoogle Scholar
  12. Curtat, L.O.: Markov equilibria of stochastic games with complementarities. Games Econ. Behav. 17, 177–199 (1996)CrossRefGoogle Scholar
  13. Das, K., Klein, N., Schmid, K.: Strategic experimentation with asymmetric players. Econ. Theory. https://doi.org/10.1007/s00199-019-01193-9 (2019)
  14. Decamps, J.-P., Mariotti, T.: Investment timing and learning externalities. J. Econ. Theory 118, 80–102 (2004)CrossRefGoogle Scholar
  15. Dutta, P.K., Rustichini, A.: A theory of stopping time games with applications to product innovations and asset sales. Econ. Theory 3, 743–763 (1993).  https://doi.org/10.1007/BF01210269 CrossRefGoogle Scholar
  16. Fudenberg, D., Tirole, J.: Preemption and rent equalization in the adoption of new technology. Rev. Econ. Stud. 52, 383–401 (1985)CrossRefGoogle Scholar
  17. Halac, M., Kartik, N., Liu, Q.: Contests for experimentation. J. Econ. Theory 125, 1523–1569 (2017)Google Scholar
  18. Heidhues, P., Rady, S., Strack, P.: Strategic experimentation with private payoffs. J. Econ. Theory 159, 531–551 (2015)CrossRefGoogle Scholar
  19. Hörner, J., Skrzypacz, A.: Learning, experimentation and information design. Working Paper, Stanford University (2016)Google Scholar
  20. Hörner, J., Klein, N.A., Rady, S.: Strongly symmetric equilibria in bandit games. Cowles Foundation Discussion Paper No. 1056. Available at SSRN. http://ssrn.com/abstract=2482335 (2014). Accessed 16 May 2019
  21. Keller, G., Rady, S.: Strategic experimentation with Poisson bandits. Theor. Econ. 5, 275–311 (2010)CrossRefGoogle Scholar
  22. Keller, G., Rady, S.: Breakdowns. Theor. Econ. 10, 175–202 (2015)CrossRefGoogle Scholar
  23. Keller, G., Rady, S., Cripps, M.: Strategic experimentation with exponential bandits. Econometrica 73, 39–68 (2005)CrossRefGoogle Scholar
  24. Khan, U., Stinchcombe, M.B.: The virtues of hesitation: optimal timing in a non-stationary world. Am. Econ. Rev. 105, 1147–1176 (2015)CrossRefGoogle Scholar
  25. Klein, N., Rady, S.: Negatively correlated bandits. Rev. Econ. Stud. 78, 693–732 (2008)CrossRefGoogle Scholar
  26. Kolmogorov, A.N., Fomin, S.V.: Introductory real analysis (Translated and edited by Silverman, R.A). Dover Books in Mathematics (1975)Google Scholar
  27. Laraki, R., Solan, E., Vieille, N.: Continuous-time games of timing. J. Econ. Theory 120, 206–238 (2005)CrossRefGoogle Scholar
  28. Laussel, D., Resende, J.: Dynamic price competition in aftermarkets with network effects. J. Math. Econ. 50, 106–118 (2014)CrossRefGoogle Scholar
  29. Marlats, C., Ménager, L.: Strategic observation with exponential bandits. Working Paper (2018)Google Scholar
  30. Pawlina, G., Kort, P.M.: Real options in asymmetric duopoly: who benefits from your comparative disadvantage? J. Econ. Manag. Strategy 15, 1–35 (2006)CrossRefGoogle Scholar
  31. Riedel, F., Steg, J.H.: Subgame-perfect equilibria in stochastic timing games. J. Math. Econ. 72, 36–50 (2017)CrossRefGoogle Scholar
  32. Rosenberg, D., Salomon, A., Vieille, N.: On games of strategic experimentation. Games Econ. Behav. 82, 31–51 (2013)CrossRefGoogle Scholar
  33. Thijssen, J.J.J., Huisman, K.J.M., Kort, P.M.: The effects of information of strategic investment and welfare. Econ. Theory 28, 399–424 (2006).  https://doi.org/10.1007/s00199-005-0628-3 CrossRefGoogle Scholar
  34. Thijssen, J.J.J., Huisman, K.J.M., Kort, P.M.: Symmetric equilibrium strategies in game theoretic real option models. J. Math. Econ. 48, 219–225 (2012)CrossRefGoogle Scholar
  35. Vives, X.: Oligopoly Pricing: Old Ideas and New Tools. MIT Press, Cambridge (1999)Google Scholar
  36. Vives, X.: Complementarities and games: new developments. J. Econ. Lit. 43, 37–479 (2005)CrossRefGoogle Scholar
  37. Vives, X.: Strategic complementarity in multi-stage games. Econ. Theory 40, 151–171 (2009).  https://doi.org/10.1007/s00199-008-0354-8 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.The University of Texas at AustinAustinUSA

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