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.

This book addresses the broad community of researchers in various fields who use relations in their scientific work. Relations occur or are used in such diverse areas as psychology, pedagogy, social choice theory, linguistics, preference modelling, ranking, multicriteria decision studies, machine learning, voting theories, spatial reasoning, data base optimization, and many more. In all these fields, and of course in mathematics and computer science, relations are used to express, to model, to reason, and to compute with. Today, problems arising in applications are increasingly handled using relational means.

In some of the above mentioned areas it sometimes looks as if the wheel is being reinvented when standard results are rediscovered in a new specialized context. Some areas are highly mathematical, others require only a moderate use of mathematics. Not all researchers are aware of the developments that relational methods have enjoyed in recent years.

A coherent text on this topic has so far not been available, and it is intended to provide one with this book. Being an overview of the field it offers an easy start, that is nevertheless theoretically sound and up to date. It will be a help to scientists even if they are not overly versed in mathematics but have to apply it. The exposition does not stress the mathematical side too early. Instead, visualizations of ideas – mainly via matrices but also via graphs – are presented first, while proofs during the early introductory chapters are postponed to an appendix.

So far we have concentrated on the foundations of relational mathematics. Now we switch to applications. A first area of applications concerns all the different variants of orderings as they originated in operations research: weakorders, semiorders, intervalorders, and block-transitive orderings. With the Scott-Suppes Theorem in relational form as well as with the study of the consecutive 1s property, we here approach research level.

The second area of applications concerns modelling preferences with relations. The hierarchy of orderings is considered investigating indifference and incomparability, often starting from so-called preference structures, i.e., relational outcomes of assessment procedures. A bibliography on early preference considerations is contained in [2].

The area of aggregating preferences with relations, studied as a third field of applications, is relatively new. It presents relational measures and integration in order to treat the trust and belief of the Dempster–Shafer Theory in relational form. In contrast, the fuzzy approach is well known, with the coeficients of matrices stemming from the real interval [0, 1]; it comes closer and closer to relational algebra proper. In the present book, a direct attempt is made. Also t-norms and De Morgan triples can be generalized to a relational form.