Stochastic games with real transition costs
and final sequence of states
Alexandru Lazari
Moldova State University, Republic of Moldova
The stochastic games with unit transition time and final sequence of
states were investigated in [2] and an efficient algorithm for determining the
game duration was developed. The purpose of this paper is to generalize
this problem for the case when the transition costs are real.
First, we do the transformation R → R+ by decreasing each transition
cost by m − 1, where m is the minimal transition cost. We have TR =
TR+ + (m − 1)T1 , where T1 , TR , TR+ are the durations of the related games.
The second step applies the rationalization schema from [1] and [3] to
do the transformation R+ → Q+ . We have TR+ ≈ TQ+ with accuracy ε > 0.
After that, we do the transformation Q+ → N∗ by changing measurement
unit. We have TQ+ = TN∗ /µ, where µ is the least common multiple of the
denominators of the rational transition costs.
Next step prepares the transition costs for extended network method
application. The function f : N∗ → N∗ , f (t) = rt + 1, is applied to each
transition cost, where r is the number of the players. We have TN∗ =
(Tf − T1 )/r, where Tf is the duration of the game with f (t) applied.
The last step applies the extended network method, described in [4]. In
this way the transformation N∗ → {1} is done. We have Tf = Text , where
Text is the game duration in given extended network. We can determine Text
using the same algorithm from [2].
References:
1. Lazari A., Lozovanu D., Capcelea M. Sisteme dinamice deterministe şi stochastice: evoluţie, optimizare şi control optimal discret. Chişinău, CEP USM, 2015,
310p.
2. Lazari A. Stationary Stochastic Games with Final Sequence of States. XIII-ème
Colloque Franco-Roumain de Mathématiques Appliquées, 2016, p. 60.
3. Lazari A. Rationalization of discrete Markov processes with final sequence states
and real transition costs. Modelare matematică, optimizare şi tehnologii informaţionale. Tezele conferinţei internaţionale. Chişinău: Evrica, 2014, p. 120-126.
4. 26. Lazari A. Extended network method for determining the main probabilistic
characteristics of the evolution for stochastic discrete systems with positive rational
transitions. Scientific abstracts of International Conference of Young Researchers,
VIII edition. Chişinău, 2010, p. 93.
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