208 research outputs found
Determinacy in Discrete-Bidding Infinite-Duration Games
In two-player games on graphs, the players move a token through a graph to
produce an infinite path, which determines the winner of the game. Such games
are central in formal methods since they model the interaction between a
non-terminating system and its environment. In bidding games the players bid
for the right to move the token: in each round, the players simultaneously
submit bids, and the higher bidder moves the token and pays the other player.
Bidding games are known to have a clean and elegant mathematical structure that
relies on the ability of the players to submit arbitrarily small bids. Many
applications, however, require a fixed granularity for the bids, which can
represent, for example, the monetary value expressed in cents. We study, for
the first time, the combination of discrete-bidding and infinite-duration
games. Our most important result proves that these games form a large
determined subclass of concurrent games, where determinacy is the strong
property that there always exists exactly one player who can guarantee winning
the game. In particular, we show that, in contrast to non-discrete bidding
games, the mechanism with which tied bids are resolved plays an important role
in discrete-bidding games. We study several natural tie-breaking mechanisms and
show that, while some do not admit determinacy, most natural mechanisms imply
determinacy for every pair of initial budgets
Obligation Blackwell Games and p-Automata
We recently introduced p-automata, automata that read discrete-time Markov
chains. We used turn-based stochastic parity games to define acceptance of
Markov chains by a subclass of p-automata. Definition of acceptance required a
cumbersome and complicated reduction to a series of turn-based stochastic
parity games. The reduction could not support acceptance by general p-automata,
which was left undefined as there was no notion of games that supported it.
Here we generalize two-player games by adding a structural acceptance
condition called obligations. Obligations are orthogonal to the linear winning
conditions that define winning. Obligations are a declaration that player 0 can
achieve a certain value from a configuration. If the obligation is met, the
value of that configuration for player 0 is 1.
One cannot define value in obligation games by the standard mechanism of
considering the measure of winning paths on a Markov chain and taking the
supremum of the infimum of all strategies. Mainly because obligations need
definition even for Markov chains and the nature of obligations has the flavor
of an infinite nesting of supremum and infimum operators. We define value via a
reduction to turn-based games similar to Martin's proof of determinacy of
Blackwell games with Borel objectives. Based on this definition, we show that
games are determined. We show that for Markov chains with Borel objectives and
obligations, and finite turn-based stochastic parity games with obligations
there exists an alternative and simpler characterization of the value function.
Based on this simpler definition we give an exponential time algorithm to
analyze finite turn-based stochastic parity games with obligations. Finally, we
show that obligation games provide the necessary framework for reasoning about
p-automata and that they generalize the previous definition
How do we remember the past in randomised strategies?
Graph games of infinite length are a natural model for open reactive
processes: one player represents the controller, trying to ensure a given
specification, and the other represents a hostile environment. The evolution of
the system depends on the decisions of both players, supplemented by chance.
In this work, we focus on the notion of randomised strategy. More
specifically, we show that three natural definitions may lead to very different
results: in the most general cases, an almost-surely winning situation may
become almost-surely losing if the player is only allowed to use a weaker
notion of strategy. In more reasonable settings, translations exist, but they
require infinite memory, even in simple cases. Finally, some traditional
problems becomes undecidable for the strongest type of strategies
Probabilistic modal {\mu}-calculus with independent product
The probabilistic modal {\mu}-calculus is a fixed-point logic designed for
expressing properties of probabilistic labeled transition systems (PLTS's). Two
equivalent semantics have been studied for this logic, both assigning to each
state a value in the interval [0,1] representing the probability that the
property expressed by the formula holds at the state. One semantics is
denotational and the other is a game semantics, specified in terms of
two-player stochastic parity games. A shortcoming of the probabilistic modal
{\mu}-calculus is the lack of expressiveness required to encode other important
temporal logics for PLTS's such as Probabilistic Computation Tree Logic (PCTL).
To address this limitation we extend the logic with a new pair of operators:
independent product and coproduct. The resulting logic, called probabilistic
modal {\mu}-calculus with independent product, can encode many properties of
interest and subsumes the qualitative fragment of PCTL. The main contribution
of this paper is the definition of an appropriate game semantics for this
extended probabilistic {\mu}-calculus. This relies on the definition of a new
class of games which generalize standard two-player stochastic (parity) games
by allowing a play to be split into concurrent subplays, each continuing their
evolution independently. Our main technical result is the equivalence of the
two semantics. The proof is carried out in ZFC set theory extended with
Martin's Axiom at an uncountable cardinal
Randomness for Free
We consider two-player zero-sum games on 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) partial-observation (both players have partial view of the game);
(b) one-sided complete-observation (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 (both players interact simultaneously); and (b) turn-based (both
players interact in turn). The two sources of randomness in these games are
randomness in transition function and randomness in strategies. In general,
randomized strategies are more powerful than deterministic strategies, and
randomness in transitions gives more general classes of games. In this work we
present a complete characterization for the classes of games where randomness
is not helpful in: (a) the transition function probabilistic transition can be
simulated by deterministic transition); and (b) strategies (pure strategies are
as powerful as randomized strategies). As consequence of our characterization
we obtain new undecidability results for these games
From Local to Global Determinacy in Concurrent Graph Games
In general, finite concurrent two-player reachability games are only determined in a weak sense: the supremum probability to win can be approached via stochastic strategies, but cannot be realized.
We introduce a class of concurrent games that are determined in a much stronger sense, and in a way, it is the largest class with this property. To this end, we introduce the notion of local interaction at a state of a graph game: it is a game form whose outcomes (i.e. a table whose entries) are the next states, which depend on the concurrent actions of the players. By definition, a game form is determined iff it always yields games that are determined via deterministic strategies when used as a local interaction in a Nature-free, one-shot reachability game. We show that if all the local interactions of a graph game with Borel objective are determined game forms, the game itself is determined: if Nature does not play, one player has a winning strategy; if Nature plays, both players have deterministic strategies that maximize the probability to win. This constitutes a clear-cut separation: either a game form behaves poorly already when used alone with basic objectives, or it behaves well even when used together with other well-behaved game forms and complex objectives.
Existing results for positional and finite-memory determinacy in turn-based games are extended this way to concurrent games with determined local interactions (CG-DLI)
Imitation in Large Games
In games with a large number of players where players may have overlapping
objectives, the analysis of stable outcomes typically depends on player types.
A special case is when a large part of the player population consists of
imitation types: that of players who imitate choice of other (optimizing)
types. Game theorists typically study the evolution of such games in dynamical
systems with imitation rules. In the setting of games of infinite duration on
finite graphs with preference orderings on outcomes for player types, we
explore the possibility of imitation as a viable strategy. In our setup, the
optimising players play bounded memory strategies and the imitators play
according to specifications given by automata. We present algorithmic results
on the eventual survival of types
A survey of stochastic ω regular games
We summarize classical and recent results about two-player games played on graphs with ω-regular objectives. These games have applications in the verification and synthesis of reactive systems. Important distinctions are whether a graph game is turn-based or concurrent; deterministic or stochastic; zero-sum or not. We cluster known results and open problems according to these classifications
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