Randomness for Free

Abstract

We consider two-player zero-sum games on finite-state graphs. These games can be classified on the basis of the information of the players and on the mode of interaction between them. On the basis of information the classification is as follows: (a) partialobservation (both players have partial view of the game); (b) one-sided completeobservation (one player has complete observation); and (c) complete-observation (both players have complete view of the game). On the basis of mode of interaction we have the following classification: (a) concurrent (players interact simultaneously); and (b) turn-based (players interact in turn). The two sources of randomness in these games are randomness in the transition function and randomness in the strategies. In general, randomized strategies are more powerful than deterministic strategies, and probabilistic transitions give more general classes of games. We present a complete characterization for the classes of games where randomness is not helpful in: (a) the transition function (probabilistic transitions can be simulated by deterministic transitions); and (b) strategies (pure strategies are as powerful as randomized strategies). As a consequence of our characterization we obtain new undecidability results for these games.

Active Diagnosis for Probabilistic Systems

“…We proceed by establishing a reduction from the problem of achieving a positive probability in a blind POMDP with a Büchi objective. This problem was shown undecidable in [CDGH10] by establishing that pure strategies are enough for POMDP. Indeed, in the case of a blind POMDP, a pure strategy is an infinite word, and so the problem boils down to the undecidable problem of the existence of an infinite word achieving a positive probability in a Büchi probabilistic automaton [BBG08].…”

“…W.l.o.g., this strategy may be assumed to be pure [CDGH10], and thus can be given as an infinite word σ = σ 1 σ 2 . .…”

“…The following immediate corollary is interesting since both the existence of a strategy achieving a Büchi objective almost surely, and the existence of strategy achieving a safety objective with positive probability are decidable for POMDP [BGG09,CDGH10]. Corollary 1.…”

Random Measurable Selections

“…This paper grew out of an attempt at proving a "randomness for free" type theorem [5] for stochastic automata [4]. We present the first steps in this direction.…”

“…This has a number of important applications, and was solved positively by Chatterjee, Doyen et al [5,Section 4], in the case that Q and L are finite. In general, we consider the following more general question: given a randomized scheduler η, and a measurable payoff function h from paths to R + , can we find two pure schedulers…”