Operator approach to values of stochastic games with varying stage duration

Sylvain Sorin and Guillaume Vigeral ()
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Guillaume Vigeral: Université Paris-Dauphine, CEREMADE

International Journal of Game Theory, 2016, vol. 45, issue 1, No 17, 389-410

Abstract: Abstract We study the links between the values of stochastic games with varying stage duration h, the corresponding Shapley operators $$\mathbf{T}$$ T and $$\mathbf{T}_h= h\mathbf{T}+ (1-h ) Id$$ T h = h T + ( 1 - h ) I d and the solution of the evolution equation $$\dot{f}_t = (\mathbf{T}- Id )f_t$$ f ˙ t = ( T - I d ) f t . Considering general non expansive maps we establish two kinds of results, under both the discounted or the finite length framework, that apply to the class of “exact” stochastic games. First, for a fixed length or discount factor, the value converges as the stage duration go to 0. Second, the asymptotic behavior of the value as the length goes to infinity, or as the discount factor goes to 0, does not depend on the stage duration. In addition, these properties imply the existence of the value of the finite length or discounted continuous time game (associated to a continuous time jointly controlled Markov process), as the limit of the value of any time discretization with vanishing mesh.

Keywords: Stochastic games; Stage duration; Shapley operator; Non expansive map; Evolution equation (search for similar items in EconPapers)
Date: 2016
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s00182-015-0512-8

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