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Time to absorption in discounted reinforcement models

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  • Pemantle, Robin
  • Skyrms, Brian
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    Abstract

    Reinforcement schemes are a class of non-Markovian stochastic processes. Their non-Markovian nature allows them to model some kind of memory of the past. One subclass of such models are those in which the past is exponentially discounted or forgotten. Often, models in this subclass have the property of becoming trapped with probability 1 in some degenerate state. While previous work has concentrated on such limit results, we concentrate here on a contrary effect, namely that the time to become trapped may increase exponentially in 1/x as the discount rate, 1-x, approaches 1. As a result, the time to become trapped may easily exceed the lifetime of the simulation or of the physical data being modeled. In such a case, the quasi-stationary behavior is more germane. We apply our results to a model of social network formation based on ternary (three-person) interactions with uniform positive reinforcement.

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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 109 (2004)
    Issue (Month): 1 (January)
    Pages: 1-12

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    Handle: RePEc:eee:spapps:v:109:y:2004:i:1:p:1-12

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    Related research

    Keywords: Network Social network Urn model Friedman urn Stochastic approximation Meta-stable Trap Three-player game Potential well Exponential time Quasi-stationary;

    References

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    1. Drew Fudenberg & David Kreps, 2010. "Learning Mixed Equilibria," Levine's Working Paper Archive 415, David K. Levine.
    2. Ellison, Glenn, 1993. "Learning, Local Interaction, and Coordination," Econometrica, Econometric Society, vol. 61(5), pages 1047-71, September.
    3. Anderlini, L. & Ianni, A., 1996. "Learning on a Torus," Discussion Paper Series In Economics And Econometrics 9611, Economics Division, School of Social Sciences, University of Southampton.
    4. A. Barrat & M. Weigt, 2000. "On the properties of small-world network models," The European Physical Journal B - Condensed Matter and Complex Systems, Springer, vol. 13(3), pages 547-560, 02.
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    Cited by:
    1. Liggett, Thomas M. & Rolles, Silke W. W., 2004. "An infinite stochastic model of social network formation," Stochastic Processes and their Applications, Elsevier, vol. 113(1), pages 65-80, September.
    2. Brian Skyrms & Robin Pemantle, 2004. "Learning to Network," Levine's Bibliography 122247000000000436, UCLA Department of Economics.
    3. Pemantle, Robin & Skyrms, Brian, 2004. "Network formation by reinforcement learning: the long and medium run," Mathematical Social Sciences, Elsevier, vol. 48(3), pages 315-327, November.

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