Competitive Markov Decision Processes by Jerzy Filar, Koos Vrieze

By Jerzy Filar, Koos Vrieze

This e-book is meant as a textual content overlaying the critical strategies and methods of aggressive Markov selection tactics. it truly is an try to current a rig­ orous therapy that mixes major learn subject matters: Stochastic video games and Markov selection tactics, which were studied exten­ sively, and now and then particularly independently, by way of mathematicians, operations researchers, engineers, and economists. given that Markov determination strategies might be considered as a different noncompeti­ tive case of stochastic video games, we introduce the hot terminology Competi­ tive Markov choice procedures that emphasizes the significance of the hyperlink among those subject matters and of the houses of the underlying Markov methods. The e-book is designed for use both in a lecture room or for self-study through a mathematically mature reader. within the advent (Chapter 1) we define a few complex undergraduate and graduate classes for which this booklet may possibly usefully function a textual content. A attribute function of aggressive Markov selection tactics - and person who encouraged our long-standing curiosity - is they can function an "orchestra" containing the "instruments" of a lot of recent utilized (and from time to time even natural) arithmetic. They represent an issue the place the tools of linear algebra, utilized likelihood, mathematical software­ ming, research, or even algebraic geometry might be "played" occasionally solo and infrequently in concord to supply both fantastically uncomplicated or both appealing, yet baroque, melodies, that's, theorems.

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Thus Xs > 0 for each s E S, and hence x has at least N positive entries. However, if x is an extreme point of X, its number of positive entries cannot exceed the rank of A. Thus we conclude that x must contain exactly one positive element in each block Xs. 40 2. Markov Decision Processes: The Noncompetitive Case (ii) In view of (i), for each s E S there exists an action as E A(s) such that 0 < x~a only if a = as. Hence for all a E A(s), s E S, fO(s, a) = x~oa = { 1, 0, Xs if a = as otherwise. This completes the proof.

This completes the proof. 5 (i) Let x be any extreme point ofX. Then each block Xs ... ,Xsm(s})T ofx contains exactly one positive element. = (Xs1,Xs2' (ii) Let xO be any basic optimal solution of the linear program (t). Then fO = M-1(xO) is a pure optimal strategy. Proof: (i) Note that constraints (i)-(iii) defining X are of the generic form b Ax x > where AT = (WT:l) is an m x (N 0, + 1) matrix. However, summing the first N rows of A yields ITW OT because for all (s,a) [1 TW] (s,a) O. The latter can be seen from the fact that N N N ~ [8(s, s') - p(s'ls, a)] ~ 8(s, s') - ~ p(s'ls, a) s'=1 8'=1 1- 1 = O.

We now shall proceed to establish the remarkable facts that not only do optimal strategies exist, but also that they correspond to optimal solutions of suitably constructed linear programs. Indeed, it will be shown that the validity of the so-called "optimality equation of dynamic programming" can be seen as a consequence of the complementary slackness theorem of linear programming. The statement of the latter theorem together with some other well-known results from linear programming can be found in Appendix G.

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