Abstract

This chapter gives an introduction to the connection between automata theory and the theory of two player games of infinite duration. We illustrate how the theory of automata on infinite words can be used to solve games with complex winning conditions, for example specified by logical formulae. Conversely, infinite games are a useful tool to solve problems for automata on infinite trees such as complementation and the emptiness test.

Introduction

The aim of this chapter is to explain some interesting connections between automata theory and games of infinite duration. The context in which these connections have been established is the problem of automatic circuit synthesis from specifications, as posed by Church [1962]. A circuit can be viewed as a device that transforms input sequences of bit vectors into output sequences of bit vectors. If the circuit acts as a kind of control device, then these sequences are assumed to be infinite because the computation should never halt.

The task in synthesis is to construct such a circuit based on a formal specification describing the desired input/output behaviour. This problem setting can be viewed as a game of infinite duration between two players: The first player provides the bit vectors for the input, and the second player produces the output bit vectors. The winning condition of the game is given by the specification. The goal is to find a strategy for the second player such that all pairs of input/output sequences that can be produced according to the strategy satisfy the specification.

Abstract

We study observation-based strategies for two-player turn-based games played on graphs with parity objectives. An observation-based strategy relies on imperfect information about the history of a play, namely, on the past sequence of observations. Such games occur in the synthesis of a controller that does not see the private state of the plant. Our main results are twofold. First, we give a fixed-point algorithm for computing the set of states from which a player can win with a deterministic observation-based strategy for a parity objective. Second, we give an algorithm for computing the set of states from which a player can win with probability 1 with a randomised observation-based strategy for a reachability objective. This set is of interest because in the absence of perfect information, randomised strategies are more powerful than deterministic ones.

Introduction

Games are natural models for reactive systems. We consider zero-sum two player turn-based games of infinite duration played on finite graphs. One player represents a control program, and the second player represents its environment. The graph describes the possible interactions of the system, and the game is of infinite duration because reactive systems are usually not expected to terminate. In the simplest setting, the game is turn-based and with perfect information, meaning that the players have full knowledge of both the game structure and the sequence of moves played by the adversary. The winning condition in a zero-sum graph game is defined by a set of plays that the first player aims to enforce, and that the second player aims to avoid.

Abstract

In this chapter we discuss relationships between logic and games, focusing on first-order logic and fixed-point logics, and on reachability and parity games. We discuss the general notion of model-checking games. While it is easily seen that the semantics of first-order logic can be captured by reachability games, more effort is required to see that parity games are the appropriate games for evaluating formulae from least fixed-point logic and the modal µ-calculus. The algorithmic consequences of this result are discussed. We also explore the reverse relationship between games and logic, namely the question of how winning regions in games are definable in logic. Finally the connections between logic and games are discussed for more complicated scenarios provided by inflationary fixed-point logic and the quantitative µ-calculus.

Introduction

The idea that logical reasoning can be seen as a dialectic game, where a proponent attempts to convince an opponent of the truth of a proposition is very old. Indeed, it can be traced back to the studies of Zeno, Socrates, and Aristotle on logic and rhetoric. Modern manifestation of this idea are the presentation of the semantics of logical formulae by means of model-checking games and the algorithmic evaluation of logical statements via the synthesis of winning strategies in such games.

model-checking games are two-player games played on an arena which is formed as the product of a structure and a formula ψ where one player, called the Verifier, attempts to prove that ψ is true in while the other player, the Falsifier, attempts to refute this.

Abstract

This is a short introduction to the subject of strategic games. We focus on the concepts of best response, Nash equilibrium, strict and weak dominance, and mixed strategies, and study the relation between these concepts in the context of the iterated elimination of strategies. Also, we discuss some variants of the original definition of a strategic game. Finally, we introduce the basics of mechanism design and use pre-Bayesian games to explain it.

Introduction

Mathematical game theory, as launched by Von Neumann and Morgenstern in their seminal book, von Neumann and Morgenstern [1944], followed by Nash's contributions Nash [1950, 1951], has become a standard tool in economics for the study and description of various economic processes, including competition, cooperation, collusion, strategic behaviour and bargaining. Since then it has also been successfully used in biology, political sciences, psychology and sociology. With the advent of the Internet game theory became increasingly relevant in computer science.

One of the main areas in game theory are strategic games (sometimes also called non-cooperative games), which form a simple model of interaction between profit maximising players. In strategic games each player has a payoff function that he aims to maximise and the value of this function depends on the decisions taken simultaneously by all players. Such a simple description is still amenable to various interpretations, depending on the assumptions about the existence of private information.

Abstract

This chapter provides an introduction to graph searching games, a form of one- or two-player games on graphs that have been studied intensively in algorithmic graph theory. The unifying idea of graph searching games is that a number of searchers wants to find a fugitive on an arena defined by a graph or hypergraph. Depending on the precise definition of moves allowed for the searchers and the fugitive and on the type of graph the game is played on, this yields a huge variety of graph searching games.

The objective of this chapter is to introduce and motivate the main concepts studied in graph searching and to demonstrate some of the central ideas developed in this area.

Introduction

Graph searching games are a form of two-player games where one player, the Searcher or Cop, tries to catch a Fugitive or Robber. The study of graph searching games dates back to the dawn of mankind: running after one another or after an animal has been one of the earliest activities of mankind and surely our hunter-gatherer ancestors thought about ways of optimising their search strategies to maximise their success.

Game playing is a powerful metaphor that fits many situations where interaction between autonomous agents plays a central role. Numerous tasks in computer science, such as design, synthesis, verification, testing, query evaluation, planning, etc. can be formulated in game-theoretic terms. Viewing them abstractly as games reveals the underlying algorithmic questions, and helps to clarify relationships between problem domains. As an organisational principle, games offer a fresh and intuitive way of thinking through complex issues.

As a result mathematical models of games play an increasingly important role in a number of scientific disciplines and, in particular, in many branches of computer science. One of the scientific communities studying and applying games in computer science has formed around the European Network ‘Games for Design and Verification’ (GAMES), which proposes a research and training programme for the design and verification of computing systems, using a methodology that is based on the interplay of finite and infinite games, mathematical logic and automata theory.

This network had initially been set up as a Marie Curie Research Training Network, funded by the European Union between 2002 and 2006. In its four years of existence this network built a strong European research community that did not exist before. Its flagship activity – the annual series of GAMES workshops – saw an ever-increasing number of participants from both within and outside Europe. The ESF Research Networking Programme GAMES, funded by the European Science Foundation ESF from 2008 to 2013, builds on the momentum of this first GAMES network, but it is scientifically broader and more ambitious, and it covers more countries and more research groups.