In this article, we start with a two-player game that models communication under adverse circumstances in everyday life and study it from the perspective of a modal logic of graphs, where links can be deleted locally according to definitions available to the adversarial player. We first introduce a new language, semantics, and some typical validities. We then formulate a new type of first-order translation for this modal logic and prove its correctness. Then, a novel notion of bisimulation is proposed which leads to a characterization theorem for the logic as a fragment of first-order logic, and a further investigation is made of its expressive power against hybrid modal languages. Next, we discuss how to axiomatize this logic of link deletion, using dynamic-epistemic logics as a contrast. Finally, we show that our new modal logic lacks both the tree model property and the finite model property, and that its satisfiability problem is undecidable.
From a board point of view, the work in this paper is an endeavor to study the interplay of modal logic with graph game. On this topic, the literature @cite_0 discusses a large number of interesting graph games in tandem with matching logics. Further more, it presents a systematical way of thinking about graph games as a meeting place with logic, and provides us inspiring ideas on research directions and questions.
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A comprehensive examination of the interfaces of logic, computer science, and game theory, drawing on twenty years of research on logic and games.This book draws on ideas from philosophical logic, computational logic, multi-agent systems, and game theory to offer a comprehensive account of logic and games viewed in two complementary ways. It examines the logic of games: the development of sophisticated modern dynamic logics that model information flow, communication, and interactive structures in games. It also examines logic as games: the idea that logical activities of reasoning and many related tasks can be viewed in the form of games.In doing so, the book takes up the ?intelligent interaction? of agents engaging in competitive or cooperative activities and examines the patterns of strategic behavior that arise. It develops modern logical systems that can analyze information-driven changes in players' knowledge and beliefs, and introduces the ?Theory of Play? that emerges from the combination of logic and game theory. This results in a new view of logic itself as an interactive rational activity based on reasoning, perception, and communication that has particular relevance for games.Logic in Games, based on a course taught by the author at Stanford University, the University of Amsterdam, and elsewhere, can be used in advanced seminars and as a resource for researchers. There has been a recent surge of interest in the role of information in strategic interactions. Much of this work seeks to understand how the realized equilibrium of a game is influenced by uncertainty in the environment and the information available to players in the game. Lurking beneath this literature is a fundamental, yet largely unexplored, algorithmic question: how should a "market maker" who is privy to additional information, and equipped with a specified objective, inform the players in the game? This is an informational analogue of the mechanism design question, and views the information structure of a game as a mathematical object to be designed, rather than an exogenous variable. We initiate a complexity-theoretic examination of the design of optimal information structures in general Bayesian games, a task often referred to as signaling. We focus on one of the simplest instantiations of the signaling question: Bayesian zero-sum games, and a principal who must choose an information structure maximizing the equilibrium payoff of one of the players. In this setting, we show that optimal signaling is computationally intractable, and in some cases hard to approximate, assuming that it is hard to recover a planted clique from an Erdos-Renyi random graph. This is despite the fact that equilibria in these games are computable in polynomial time, and therefore suggests that the hardness of optimal signaling is a distinct phenomenon from the hardness of equilibrium computation. Necessitated by the non-local nature of information structures, en-route to our results we prove an "amplification lemma" for the planted clique problem which may be of independent interest. Specifically, we show that even if we plant many cliques in an Erdos-Renyi random graph, so much so that most nodes in the graph are in some planted clique, recovering a constant fraction of the planted cliques is no easier than the traditional planted clique problem. The author discusses games of both perfect and imperfect information at two levels of structural detail: players' local actions, and their global powers for determining outcomes of the game. Matching logical languages are proposed for both. In particular, at the "action level", imperfect information games naturally model a combined "dynamic-epistemic language"--and correspondences are found between special axioms in this language and particular modes of playing games with their information dynamics. At the "outcome level", the paper presents suitable notions of game equivalence, and some simple representation results. Copyright 2001 by Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of Economic Research Consider the following probabilistic one-player game: The board is a graph with @math vertices, which initially contains no edges. In each step, a new edge is drawn uniformly at random from all nonedges and is presented to the player, henceforth called Painter. Painter must assign one of @math available colors to each edge immediately, where @math is a fixed integer. The game is over as soon as a monochromatic copy of some fixed graph @math has been created, and Painter's goal is to “survive” for as many steps as possible before this happens. We present a new technique for deriving upper bounds on the threshold of this game, i.e., on the typical number of steps Painter will survive with an optimal strategy. More specifically, we consider a deterministic two-player variant of the game where the edges are chosen not randomly, but by a second player Builder. However, Builder has to adhere to the restriction that, for some real number @math , the ratio of edges to vertices in all subgraphs of the evolving board ne...