Competitive Markov Decision Processes - download pdf or read online

By Jerzy Filar, Koos Vrieze

ISBN-10: 1461240549

ISBN-13: 9781461240549

ISBN-10: 1461284813

ISBN-13: 9781461284819

This publication is meant as a textual content overlaying the valuable recommendations and strategies of aggressive Markov choice methods. it's an try and current a rig­ orous therapy that mixes major examine issues: Stochastic video games and Markov selection methods, which were studied exten­ sively, and now and then relatively independently, via mathematicians, operations researchers, engineers, and economists. considering the fact that Markov selection methods will be seen as a distinct noncompeti­ tive case of stochastic video games, we introduce the recent terminology Competi­ tive Markov determination tactics that emphasizes the significance of the hyperlink among those themes and of the homes of the underlying Markov approaches. The e-book is designed for use both in a school room or for self-study by way of a mathematically mature reader. within the creation (Chapter 1) we define a couple of complicated undergraduate and graduate classes for which this ebook might usefully function a textual content. A attribute characteristic of aggressive Markov determination 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 matter the place the tools of linear algebra, utilized likelihood, mathematical software­ ming, research, or even algebraic geometry will be "played" occasionally solo and infrequently in concord to supply both superbly basic or both attractive, yet baroque, melodies, that's, theorems.

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1 that the stationary distribution q(f) induced by f E F s satisfies the linear system of equations q(f) [I - P(f)] = O. 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.

We now shall derive a useful partition of the class F D of deterministic strategies that is based on the graphs they "trace out" in G. In particular, note that with each f E F D we can associate a subgraph Gf of G defined by arc (s, s') E G f ~ f(s) = s'. We also shall denote a simple cycle of length m and beginning at 1 by a set of arcs Of course, c~ is a Hamiltonian cycle. If G f contains a cycle c;", we write G f ::J c;,.. Let C m := {f E FDIG f ::J c;"}, namely, the set of deterministic strategies that trace out a simple cycle of length m, beginning at 1, for each m = 2,3, ...

1 (continued) With the data as given earlier there are only two pure stationary policies in this model: f1 = (( 1, 0) ( 1)) and f2 = (( 0, 1) (1)) . Starting from state 1 it is not immediately clear which one of f1 or f2 is better. Of course, from state 2 both f1 and f2 yield -5 --=-50. 9 Let us conjecture that action 2 in state 1 is better than action 1, that is, that the higher probability of remaining in state 1 more than compensates for the lower reward resulting from action 2. In such a case f2 will be the optimal control and Vi3(f2) will be the value vector that should satisfy the optimality equation.

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Competitive Markov Decision Processes by Jerzy Filar, Koos Vrieze

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