Positional Determinacy of Games with Infinitely Many Priorities

We study two-player games of infinite duration that are played on finite or infinite game graphs. A winning strategy for such a game is positional if it only depends on the current position, and not on the history of the play. A game is positionally determined if, from each position, one of the two players has a positional winning strategy. The theory of such games is well studied for winning conditions that are defined in terms of a mapping that assigns to each position a priority from a finite set. Specifically, in Muller games the winner of a play is determined by the set of those priorities that have been seen infinitely often; an important special case are parity games where the least (or greatest) priority occurring infinitely often determines the winner. It is well-known that parity games are positionally determined whereas Muller games are determined via finite-memory strategies. In this paper, we extend this theory to the case of games with infinitely many priorities. Such games arise in several application areas, for instance in pushdown games with winning conditions depending on stack contents. For parity games there are several generalisations to the case of infinitely many priorities. While max-parity games over omega or min-parity games over larger ordinals than omega require strategies with infinite memory, we can prove that min-parity games with priorities in omega are positionally determined. Indeed, it turns out that the min-parity condition over omega is the only infinitary Muller condition that guarantees positional determinacy on all game graphs

On Modal {\mu}-Calculus over Finite Graphs with Bounded Strongly Connected Components

For every positive integer k we consider the class SCCk of all finite graphs whose strongly connected components have size at most k. We show that for every k, the Modal mu-Calculus fixpoint hierarchy on SCCk collapses to the level Delta2, but not to Comp(Sigma1,Pi1) (compositions of formulas of level Sigma1 and Pi1). This contrasts with the class of all graphs, where Delta2=Comp(Sigma1,Pi1)

The Fixpoint-Iteration Algorithm for Parity Games

It is known that the model checking problem for the modal mu-calculus reduces to the problem of solving a parity game and vice-versa. The latter is realised by the Walukiewicz formulas which are satisfied by a node in a parity game iff player 0 wins the game from this node. Thus, they define her winning region, and any model checking algorithm for the modal mu-calculus, suitably specialised to the Walukiewicz formulas, yields an algorithm for solving parity games. In this paper we study the effect of employing the most straight-forward mu-calculus model checking algorithm: fixpoint iteration. This is also one of the few algorithms, if not the only one, that were not originally devised for parity game solving already. While an empirical study quickly shows that this does not yield an algorithm that works well in practice, it is interesting from a theoretical point for two reasons: first, it is exponential on virtually all families of games that were designed as lower bounds for very particular algorithms suggesting that fixpoint iteration is connected to all those. Second, fixpoint iteration does not compute positional winning strategies. Note that the Walukiewicz formulas only define winning regions; some additional work is needed in order to make this algorithm compute winning strategies. We show that these are particular exponential-space strategies which we call eventually-positional, and we show how positional ones can be extracted from them.Comment: In Proceedings GandALF 2014, arXiv:1408.556

Playing Games in the Baire Space

We solve a generalized version of Church's Synthesis Problem where a play is given by a sequence of natural numbers rather than a sequence of bits; so a play is an element of the Baire space rather than of the Cantor space. Two players Input and Output choose natural numbers in alternation to generate a play. We present a natural model of automata ("N-memory automata") equipped with the parity acceptance condition, and we introduce also the corresponding model of "N-memory transducers". We show that solvability of games specified by N-memory automata (i.e., existence of a winning strategy for player Output) is decidable, and that in this case an N-memory transducer can be constructed that implements a winning strategy for player Output.Comment: In Proceedings Cassting'16/SynCoP'16, arXiv:1608.0017

Limit Your Consumption! Finding Bounds in Average-energy Games

Energy games are infinite two-player games played in weighted arenas with quantitative objectives that restrict the consumption of a resource modeled by the weights, e.g., a battery that is charged and drained. Typically, upper and/or lower bounds on the battery capacity are part of the problem description. Here, we consider the problem of determining upper bounds on the average accumulated energy or on the capacity while satisfying a given lower bound, i.e., we do not determine whether a given bound is sufficient to meet the specification, but if there exists a sufficient bound to meet it. In the classical setting with positive and negative weights, we show that the problem of determining the existence of a sufficient bound on the long-run average accumulated energy can be solved in doubly-exponential time. Then, we consider recharge games: here, all weights are negative, but there are recharge edges that recharge the energy to some fixed capacity. We show that bounding the long-run average energy in such games is complete for exponential time. Then, we consider the existential version of the problem, which turns out to be solvable in polynomial time: here, we ask whether there is a recharge capacity that allows the system player to win the game. We conclude by studying tradeoffs between the memory needed to implement strategies and the bounds they realize. We give an example showing that memory can be traded for bounds and vice versa. Also, we show that increasing the capacity allows to lower the average accumulated energy.Comment: In Proceedings QAPL'16, arXiv:1610.0769

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)

Extending finite-memory determinacy to multi-player games

We show that under some general conditions the finite memory determinacy of a class of two-player win/lose games played on finite graphs implies the existence of a Nash equilibrium built from finite memory strategies for the corresponding class of multi-player multi-outcome games. This generalizes a previous result by Brihaye, De Pril and Schewe. We provide a number of example that separate the various criteria we explore.Our proofs are generally constructive, that is, provide upper bounds for the memory required, as well as algorithms to compute the relevant Nash equilibria