Download Competitive Markov Decision Processes by Jerzy Filar, Koos Vrieze PDF

By Jerzy Filar, Koos Vrieze

This publication is meant as a textual content protecting the crucial ideas and strategies of aggressive Markov determination strategies. it truly is an try to current a rig­ orous therapy that mixes major examine issues: Stochastic video games and Markov determination strategies, which were studied exten­ sively, and every now and then relatively independently, via mathematicians, operations researchers, engineers, and economists. on account that Markov determination methods could be seen as a unique noncompeti­ tive case of stochastic video games, we introduce the hot terminology Competi­ tive Markov choice strategies that emphasizes the significance of the hyperlink among those subject matters and of the homes of the underlying Markov strategies. The publication is designed for use both in a lecture room or for self-study by means of a mathematically mature reader. within the advent (Chapter 1) we define a few complicated undergraduate and graduate classes for which this ebook may well usefully function a textual content. A attribute function of aggressive Markov selection methods - and person who encouraged our long-standing curiosity - is they can function an "orchestra" containing the "instruments" of a lot of recent utilized (and now and then even natural) arithmetic. They represent a subject the place the tools of linear algebra, utilized likelihood, mathematical software­ ming, research, or even algebraic geometry might be "played" occasionally solo and occasionally in concord to provide both superbly easy or both appealing, yet baroque, melodies, that's, theorems.

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However, the above vector equation can be written term by term as N L (8(s, s') - p(s'ls, f)) qs(f) = 0, 8=1 1 Recall that q is a row of the Cesaro-limit matrix Q. 4 The Irreducible Limiting Average Process 35 where 6(s, s') is the Kronecker delta. 26)) N L L (6(s, s') - p(s'ls, a)) qs(f)f(s, a) s=l aEA(s) N = L L (6(s, s') - p(s'ls, a)) xsa(f) = 0; s' E S. s=l aEA(s) Further, since N L L N xsa(f) = N L L qs(f)f(s, a) = L qs(f) = 1, s=l s=l aEA(s) s=l aEA(s) we naturally are led to consider the polyhedral set X defined by the linear constraints N (i) L L (6(s, s') - p(s'ls, a)) Xsa = 0, s' E S s=l aEA(s) N (ii) L L Xsa = 1 s=l aEA(s) (iii) Xsa :::: 0; a E A(s), s E S.

Let r (3, r T, and r a denote the discounted, terminating, and limiting average models considered so far. Then there is no loss of generality in restricting analysis to F M since and (ii) sUPV{3(s,7r) FB = supv{3(s, 7r), supvT(s, 7r) = supvT(s, 7r) FM and supva(s,7r) FB = FB FM supva(s, 7r). FM Furthermore, (i) and (ii) hold for any other performance criterion that aggregates the sequence {lEs,7r[Rt J}:o; 7r E F B , S E S. Proof: Note that each of the performance criteria mentioned above aggregates the sequence of expected rewards/outputs.

2 Fix an arbitary S E S. 36). Let r (3, r T, and r a denote the discounted, terminating, and limiting average models considered so far. Then there is no loss of generality in restricting analysis to F M since and (ii) sUPV{3(s,7r) FB = supv{3(s, 7r), supvT(s, 7r) = supvT(s, 7r) FM and supva(s,7r) FB = FB FM supva(s, 7r). FM Furthermore, (i) and (ii) hold for any other performance criterion that aggregates the sequence {lEs,7r[Rt J}:o; 7r E F B , S E S. Proof: Note that each of the performance criteria mentioned above aggregates the sequence of expected rewards/outputs.

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