Abstract. Tree-width is a well-known metric on undirected graphs that measures how tree-like a graph is and gives a notion of graph decomposition that proves useful in algorithm development. Tree-width is characterised by a game known as the cops-and-robber game where a number of cops chase a robber on the graph. We consider the natural adaptation of this game to directed graphs and show that monotone strategies in the game yield a measure with an associated notion of graph decomposition that can be seen to describe how close a directed graph is to a directed acyclic graph (DAG). This promises to be useful in developing algorithms on directed graphs. In particular, we show that the problem of determining the winner of a parity game is solvable in polynomial time on graphs of bounded DAG-width. We also consider the relationship between DAG-width and other measures of such as entanglement and directed tree-width. One consequence we obtain is that certain NP-complete problems such as Hamiltonicity and disjoint paths are polynomial-time computable on graphs of bounded DAG-width.

We analyse the notion of iterated admissibility, i.e., avoidance of weakly dominated strategies, as a solution concept for extensive games of infinite horizon. This concept is known to provide a valuable criterion for selecting among multiple equilibria and to yield sharp predictions in finite games. However, generalisations to the infinite are inherently problematic, due to unbounded dominance chains and the requirement of transfinite induction. In a multi-player non-zero-sum setting, we show that for infinite extensive games of perfect information with only two possible payoffs (win or lose), the concept of iterated admissibility is sound and robust: all iteration stages are dominated by admissible strategies, the iteration is non-stagnating, and, under regular winning conditions, strategies that survive iterated elimination of dominated strategies form a regular set. This research was supported by the eu rtn 'games' (www.games.rwth-aachen.de).

Tree-width is a well-known metric on undirected graphs that measures how tree-like a graph is and gives a notion of graph decomposition that proves useful in algorithm design. Tree-width can be characterised by a graph searching game where a number of cops attempt to capture a robber. We consider the natural adaptation of this game to directed graphs and show that monotone strategies in the game yield a measure, called dag-width, that can be seen to describe how close a directed graph is to a directed acyclic graph (dag). We also provide an associated decomposition and show how it is useful for developing algorithms on directed graphs. In particular, we show that the problem of determining the winner of a parity game is solvable in polynomial time on graphs of bounded dag-width. We also consider the relationship between dag-width and other connectivity measures such as directed tree-width and path-width. A consequence we obtain is that certain NP-complete problems such as Hamiltonicity and disjoint paths are polynomialtime computable on graphs of bounded dag-width.

We propose a new parameter for the complexity of finite directed graphs which measures to what extent the cycles of the graph are intertwined. This measure, called entanglement, is defined by way of a game that is somewhat similar in spirit to the robber and cops games used to describe tree width, directed tree width, and hypertree width. Nevertheless, on many classes of graphs, there are significant differences between entanglement and the various incarnations of tree width. Entanglement is intimately connected to the computational and descriptive complexity of the modal µ-calculus. On the one hand, the number of fixed point variables needed to describe a finite graph up to bisimulation is captured by its entanglement. This plays a crucial role in the proof that the variable hierarchy of the µ-calculus is strict. In addition to this, we prove that parity games of bounded entanglement can be solved in polynomial time. Specifically, we establish that the complexity of solving a parity game can be parametrised in terms of the minimal entanglement of a subgame induced by a winning strategy.

We consider imperfect-information parity games in which strategies rely on observations that provide imperfect information about the history of a play. To solve such games, i.e., to determine the winning regions of players and corresponding winning strategies, one can use the subset construction to build an equivalent perfect-information game. Recently, an algorithm that avoids the inefficient subset construction has been proposed. The algorithm performs a fixed-point computation in a lattice of antichains, thus maintaining a succinct representation of state sets. However, this representation does not allow to recover winning strategies. In this paper, we build on the antichain approach to develop an algorithm for constructing the winning strategies in parity games of imperfect information. We have implemented this algorithm as a prototype. To our knowledge, this is the first implementation of a procedure for solving imperfect-information parity games on graphs.

When seeking to coordinate in a game with imperfect information, it is often relevant for a player to know what other players know. Keeping track of the information acquired in a play of infinite duration may, however, lead to infinite hierarchies of higher-order knowledge. We present a construction that makes explicit which higher-order knowledge is relevant in a game and allows us to describe a class of games that admit coordinated winning strategies with finite memory.

Most of the logics commonly used in verification, such as LTL, CTL, CTL * , and PDL can be embedded into the two-variable fragment of the µ-calculus. It is also known that properties occurring at arbitrarily high levels of the alternation hierarchy can be formalised using only two variables. This raises the question whether the number of fixedpoint variables in µ-formulae can be bounded in general.We answer this question negatively, and prove that the variablehierarchy of the µ-calculus is semantically strict. For any k, we provide examples of formulae with k variables that are not equivalent to any formula with fewer variables. In particular, this implies that Parikh's Game Logic is less expressive than the µ-calculus, thus resolving an open issue raised by Parikh in 1983.

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