Chapter 1 Logic and Game Theory (original) (raw)
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On modal logic interpretations of games
2002
Multi-agent environments comprise decision makers whose deliberations involve reasoning about the expected behavior of other agents. Apposite concepts of rational choice have been studied and formalized in game theory and our particular interest is with their integration in a logical specification language for multiagent systems. This paper concerns the logical analysis of the gametheoretical notions of a (subgame perfect) Nash equilibrium and that of a (subgame perfect) best response strategy. Extensive forms of games are conceived of as Kripke frames and a version of Propositional Dynamic Logic is employed to describe them. We show how formula schemes of our language characterize those classes of frames in which the strategic choices of the agents can be said to be Nashoptimal. Our analysis focuses on extensive games of perfect information without repetition.
Epistemic Game Theory and Logic: Introduction
Games
Epistemic game theory and the systems of logic that support it are crucial for understanding rational behavior in interactive situations in which the outcome for an agent depends, not just on her own behavior, but also on the behavior of those with whom she is interacting. Scholars in many fields study such interactive situations, that is, games of strategy. Epistemic game theory presents the epistemic foundations of a game's solution, taken as a combination of strategies, one for each player in the game, such that each strategy is rational given the combination. It considers the beliefs of the players in a game and shows how, along with the players' goals, their beliefs guide their choices and settle the outcome of their game. Adopting the Bayesian account of probability, as rational degree of belief, it yields Bayesian game theory. Epistemic game theory, because it attends to how players reason strategically in games, contrasts with evolutionary game theory, which applies to non-reasoning organisms such as bacteria. Logic advances rules of inference for strategic reasoning. It contributes not just standard rules of deductive logic, such as modus ponens, but also rules of epistemic logic, such as the rule going from knowledge of a set of propositions to knowledge of their deductive consequences, and rules of probabilistic reasoning such as Bayesian conditionalization, which uses probabilities conditional on receiving some new evidence to form new non-conditional probabilities after receiving exactly that new evidence. Perea [1] offers an overview, and Weirich [2] shows how principles of choice support solutions to games of strategy. The papers in the special issue came in response to the journal's call for papers. Diversity of perspectives was a goal. The papers include four by economists, one by computer scientists, three by philosophers, and one by a psychologist. They display a variety of approaches to epistemic game theory and logic. The following paragraphs briefly describe the topics of the papers, grouped according to discipline and within a discipline according to date of publication.
Modal logic and game theory: two alternative approaches
Risk Decision and Policy, 2002
Two views of game theory are discussed: (1) game theory as a description of the behavior of rational individuals who recognize each other's reationality and reasoning abilities, and (2) game theory as an internally consistent recommendation to individuals on how to act in interactive situations. It is shown that the same mathematical tool, namely modal logic, can be used to explicitly model both views.
Epistemic Game Theory and Logic
MDPI eBooks, 2017
Preface to "Epistemic Game Theory and Logic" Epistemic game theory and the systems of logic that support it are crucial for understanding rational behavior in interactive situations in which the outcome for an agent depends, not just on her own behavior, but also on the behavior of those with whom she is interacting. Scholars in many fields study such interactive situations, that is, games of strategy. Epistemic game theory presents the epistemic foundations of a game's solution, taken as a combination of strategies, one for each player in the game, such that each strategy is rational given the combination. It considers the beliefs of the players in a game and shows how, along with the players' goals, their beliefs guide their choices and settle the outcome of their game. Adopting the Bayesian account of probability, as rational degree of belief, it yields Bayesian game theory. Epistemic game theory, because it attends to how players reason strategically in games, contrasts with evolutionary game theory, which applies to non-reasoning organisms such as bacteria. Logic advances rules of inference for strategic reasoning. It contributes not just standard rules of deductive logic, such as modus ponens, but also rules of epistemic logic, such as the rule going from knowledge of a set of propositions to knowledge of their deductive consequences, and rules of probabilistic reasoning such as Bayesian conditionalization, which uses probabilities conditional on receiving some new evidence to form new non-conditional probabilities after receiving exactly that new evidence. Perea (2012) offers an overview, and Weirich (1998) shows how principles of choice support solutions to games of strategy. The papers in the special issue came in response to the journal's call for papers. Diversity of perspectives was a goal. The papers include four by economists, one by computer scientists, three by philosophers, and one by a psychologist. They display a variety of approaches to epistemic game theory and logic. The following paragraphs briefly describe the topics of the papers, grouped according to discipline and within a discipline according to date of publication.
Epistemic logics and their game theoretic applications: Introduction
Economic Theory, 2002
This paper is written as an introduction to epistemic logics and their game theoretic applications. It starts with both semantics and syntax of classical logic, and goes to the Hilbert-style proof-theory and Kripke-style model theory of epistemic logics. In these theories, we discuss individual decision making in some simple game examples. In particular, we will discuss the distinction between beliefs and knowledge, and how false beliefs play roles in game theoretic decision making. Finally, we discuss extensions of epistemic logics to incorporate common knowledge. In the extension, we discuss also false beliefs on common knowledge.
Revisiting games in dynamic-epistemic logic
2018
We revisit the discussion on reasoning about games in dynamic-epistemic logic and present a language for describing reasoning in possibly infinite games from the perspective of the players. We argue that even though a plethora of sophisticated logics of strategic reasoning in games are available, it is still worthwhile to consider the game structures themselves from the perspective of logic. In the process, we provide complete axiom systems for these games satisfying characteristic properties from the gametheoretic literature. Decidability of the satisfiability problem is also taken up to consider the existence of games following certain rules that can be expressed in the logical language.
Erratum: "Rational Dynamics and Epistemic Logic in Games
International Game Theory Review, 2007
Game-theoretic solution concepts describe sets of strategy profiles that are optimal for all players in some plausible sense. Such sets are often found by recursive algorithms like iterated removal of strictly dominated strategies in strategic games, or backward induction in extensive games. Standard logical analyses of solution sets use assumptions about players in fixed epistemic models for a given game, such as mutual knowledge of rationality. In this paper, we propose a different perspective, analyzing solution algorithms as processes of learning which change game models. Thus, strategic equilibrium gets linked to fixed-points of operations of repeated announcement of suitable epistemic statements. This dynamic stance provides a new look at the current interface of games, logic, and computation.
The B.E. Journal of Theoretical Economics, 2018
This special section of the B.E. Journal of Theoretical Economics contains a selection of papers presented at the 12 ℎ Conference on Logic and the Foundations of Game and Decision Theory (LOFT12), which took place in Maastricht (The Netherlands), July 20-22, 2016. While this special section collects papers that have a stronger epistemological and game-theoretic content, a second set of papers-which are more focused on logic-can be found in a companion special issue of Studia Logica. The LOFT conferences have spanned a period of 22 years: the first took place in 1994 in Marseille (France) and, since then, LOFT has become a regular biannual event. 1 The LOFT conferences are interdisciplinary events that bring together researchers from a variety of fields: cognitive psychology, computer science and artificial intelligence, economics, game theory, linguistics, logic, mind sciences, philosophy and social choice. In its original conception, LOFT had as its central theme the application of logic, in particular modal epistemic logic, to foundational issues in the theory of games and individual decision-making. The LOFT initiative arose from the realization that the tools and methodology that were used in game theory were closely related to those used in other fields, notably computer science, logic and philosophy. Modal logic turned out to be the common language that made it possible to bring together different professional communities. New and active areas of research have sprung from the interdisciplinary exposure provided by the LOFT events. Over time the scope of the LOFT conferences has broadened to encompass a wider range of topics, while maintaining its focus on the general issue of rationality and agency. Topics that have fallen within the LOFT umbrella include epistemic and temporal logic, theories of information processing and belief revision, models of bounded rationality, non-monotonic reasoning, theories of learning and evolution, social choice theory, the theory of social networks, etc. A complete list of publications that have sprung from the past LOFT conferences is given in the References section at the end of this Introduction. This special section consists of five articles, which are briefly summarized below. Theory of mind is the topic of "Estimating the use of higher-order theory of mind using computational agents" by Harmen de Weerd, Denny Diepgrond and Rineke Verbrugge. The authors use a combination of computational agents and Bayesian model selection to determine to what extent people make use of higherorder theory of mind reasoning in a particular competitive game known as matching pennies. This method allows them to consider a population with differences among individuals in use of theory of mind, in contrast to the existing estimation methods in the behavioral economics literature, which determine which level of theory of mind reasoning best describes the population as a whole. The paper's goal is to test the effectiveness of the Bayesian estimation procedure, as well as to determine to what extent human participants make use of theory of mind in simple competitive games. To do so, they apply this method to two empirical studies in which participants play the matching pennies game. They find that while many children and adults appear to make use of theory of mind, participants are also often classified as using a simpler strategy based only on the actions of the directly preceding round. This may indicate that human reasoners do not primarily use their theory of mind abilities to compete with others. The paper "Beyond coincidence: the reasoning process underlying utility proportional beliefs process" by Christian Tobias Nauerz provides new insights into a solution concept for strategic-form games introduced by Bach and Perea (2014), namely the concept of "utility proportional beliefs" (UPB). The main idea behind