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Time to absorption in discounted reinforcement models

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  • Pemantle, Robin
  • Skyrms, Brian

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.

Suggested Citation

  • Pemantle, Robin & Skyrms, Brian, 2004. "Time to absorption in discounted reinforcement models," Stochastic Processes and their Applications, Elsevier, vol. 109(1), pages 1-12, January.
    • Handle: RePEc:eee:spapps:v:109:y:2004:i:1:p:1-12
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    References listed on IDEAS

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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. José Moler & Fernando Plo & Henar Urmeneta, 2013. "A generalized Pólya urn and limit laws for the number of outputs in a family of random circuits," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(1), pages 46-61, March.
    4. 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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