By Jerzy Filar, Koos Vrieze

This booklet is meant as a textual content masking the significant innovations and strategies of aggressive Markov determination procedures. it truly is an try and current a rig orous remedy that mixes major examine issues: Stochastic video games and Markov selection approaches, which were studied exten sively, and from time to time rather independently, through mathematicians, operations researchers, engineers, and economists. on account that Markov choice techniques will be considered as a distinct noncompeti tive case of stochastic video games, we introduce the recent terminology Competi tive Markov determination techniques that emphasizes the significance of the hyperlink among those themes and of the homes of the underlying Markov techniques. The ebook is designed for use both in a school room or for self-study via a mathematically mature reader. within the advent (Chapter 1) we define a couple of complex undergraduate and graduate classes for which this booklet might usefully function a textual content. A attribute function of aggressive Markov choice strategies - and one who encouraged our long-standing curiosity - is they can function an "orchestra" containing the "instruments" of a lot of recent utilized (and every now and then even natural) arithmetic. They represent a subject the place the tools of linear algebra, utilized likelihood, mathematical application ming, research, or even algebraic geometry may be "played" occasionally solo and infrequently in concord to provide both fantastically easy or both attractive, yet baroque, melodies, that's, theorems.

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**Additional info for Competitive Markov Decision Processes**

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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. 2 Let S = {1, 2}, A(1) := 6(s, s') - p(s'ls, a). 3) state 2 36 2. Markov Decision Processes: The Noncompetitive Case The reader is invited to verify the fact that this limiting average model is indeed irreducible.

P = 1) results in a deterministic stream of outputs 10, -8, 10, -8, ... when starting in state 1. The corresponding limiting average value is vo (l, f(l)) = 1. 4 The Irreducible Limiting Average Process 33 On the other hand, f(p) with p < 1 will result in an eventual absorption in state 3. Hence, even though f(p) with p close or equal to 0 can be expected to produce a long sequence of outputs of 20, its limiting average output will be the dismal v o (1, f(p)) = O! The above heuristic argument can be made precise with the help of the "zero-one law" of probability.

1. Then 71"* is an optimal strategy over F~, and for all n = 0,1, ... , T and s E S Vn(s) = max {r(s, a) A(8) + t p(s'ls, a)Vn-1(S')} . 1. We now need to prove the optimality of 71"*. For n = 0 and an arbitrary 71" = (10, II, ... , IT) E FE, it follows immediately from Step 1 of the algorithm that for all s' E S IE-rr [RTIST = s'] < max {r(s',a)} Acs') IE-rr" [RTIST = s'] Vo(s'). 10) for all s' E S. Now consider IE-rr [t=~n R t IST-n sIll = L aEACs") IE-rr [ t t=T-n R t I ST-n = s", A T- n = al fT-n(s", a).