Acta Mathematicae Applicatae Sinica

, Volume 3, Issue 1, pp 15–25 | Cite as

On the properties of ε(≥0) optimal policies in discounted unbounded return model

  • Dong Zeqing 
  • Zhang Sheng 
Article

Abstract

This paper investigates the properties of ε(≥0) optimal policies in the model of [2]. It is shown that, if π* = (π 0 * , π 1 * ,..., π n * , π n +1/* , ...) is aβ-discounted optimal policy, then (π 0 * , π 1 * , ..., π n * ) for alln≥0 is also aβ-discounted optimal policy. Under some condition we prove that stochastic stationary policy π n *∞ corresponding to the decision rule π n * is also optimal for the same discounting factorβ. We have also shown that for eachβ-optimal stochastic stationary policy π 0 *∞ , π 0 *∞ can be decomposed into several decision rules to which the corresponding stationary policies are alsoβ-optimal separately; and conversely, a proper convex combination of these decision rules is identified with the former π 0 * . We have further proved that for any (ε,β)-optimal policy, say π*=(π 0 * , π 1 * , ..., π n * , π n +1/* , ...), π n−1 * ) is ((1−β n )−1e, β) optimal forn>0. At the end of this paper we mention that the results about convex combinations and decompositions of optimal policies of § 4 in [1] can be extended to our case.

Keywords

Decision Rule Stationary Policy Optimal Policy Convex Combination Math Application 

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References

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Copyright information

© Science Press, Beijing, China and Allerton Press, Inc. New York, U.S.A. 1987

Authors and Affiliations

  • Dong Zeqing 
    • 1
  • Zhang Sheng 
    • 2
  1. 1.Institute of Applied MathematicsAcademia SinicaChina
  2. 2.Yunnan UniversityChina

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