Common  Knowledge in Game Theory

Abstract Decision theory and game theory are extended to allow for information processing errors. This extended theory is then used to reexamine market speculation and consensus, both when all actions (opinions) are common knowledge and when they may not be. Five axioms of information processing are shown to be especially important to speculation and consensus. They are called nondelusion, knowing that you know (KTYK), nested, balanced, and positively balanced. We show that it is necessary and sufficient that each agent's information processing errors be (1) nondeluded and balanced so that the agents cannot agree to disagree, (2) nondeluded and positively balanced so that it cannot be common knowledge that they are speculating, and (3) nondeluded and KTYK and nested so that agents cannot speculate in equilibrium. Each condition is strictly weaker than the next one, and the last is strictly weaker than partition information.

Download Full-text

Game Theory and the Law

Theories of Choice ◽

10.1093/oso/9780198863175.003.0004 ◽

2021 ◽

pp. 55-70

Author(s):

Marlies Ahlert

Keyword(s):

Game Theory ◽

Common Knowledge ◽

Ideal Theory ◽

Specific Information ◽

Strategic Interactions ◽

Bargaining Problems ◽

Game Forms ◽

Classical Game ◽

Micro Foundations ◽

Adaptation Model

Classical game theory analyses strategic interactions under extreme idealisations. It assumes cognitively unconstrained players with common knowledge concerning game forms, preferences, and rationality. Such ideal theory is highly relevant for human self-understanding as a rational being or what Selten called ‘rationology’. Yet, ideal theory is highly irrelevant for real actors who are in Selten’s sense boundedly rational. Starting from essential features of real bargaining problems, elements of Selten’s ‘micro-psychological’ and Raiffa’s ‘telescopic’ behavioural bargaining theory are introduced. From this, an outline of a workable rationality approach to bargaining emerges. It suggests relying on telescopic elements from Raiffa’s model to provide general outcome orientation and on insights from Selten’s aspiration adaptation model of individual decision making to develop process-sensitive action advice. A bird’s eye view of a prominent recent case of ‘bargaining in the shadow of the courts’ shows a surprisingly good fit of outcomes with the implications of Raiffa’s telescopic approach while remaining compatible with a Seltenian process. Though due to a lack of specific information because the micro-foundations for the telescopic theory cannot be provided, it is at least clear how further case studies and experiments might be put to work here.

Download Full-text

Common Knowledge and Cheap Talk in Democratic Discourse and Law

10.1093/oso/9780190084486.003.0006 ◽

2021 ◽

pp. 147-164

Author(s):

Richard R. W. Brooks

Keyword(s):

Game Theory ◽

Common Knowledge ◽

Common Law ◽

Cheap Talk ◽

Public Communication ◽

Costly Signaling ◽

The Public ◽

Game Theoretic ◽

Key Aspects

This commentary illuminates key aspects of Shiffrin’s view by appeal to concrete examples and notions from game theory. It underscores the role of law as a means for the public communication of moral commitments by invoking the idea of common knowledge. Our commitments must be known to be shared, that knowledge itself must be known to be shared, and so on ad infinitum. This offers a perspective on the importance of common law from a democratic framework: common law can be seen as a mechanism for generating common knowledge about disputes and their resolution. The commentary invokes another game-theoretic notion, that of the contrast between cheap talk and costly signaling, to illuminate Shiffrin’s discussion of constitutional balancing. Where the interests of speaker and addressee are not aligned, cheap talk lacks credibility, and this is something to which courts need to be sensitive in balancing state and constitutional interests.

Download Full-text

COMMON KNOWLEDGE, SALIENCE AND CONVENTION: A RECONSTRUCTION OF DAVID LEWIS' GAME THEORY

Economics and Philosophy ◽

10.1017/s0266267103001123 ◽

2003 ◽

Vol 19 (2) ◽

pp. 175-210 ◽

Cited By ~ 81

Author(s):

Robin P. Cubitt ◽

Robert Sugden

Keyword(s):

Game Theory ◽

Common Knowledge ◽

Inductive Reasoning ◽

David Lewis ◽

Rigorous Analysis ◽

Explicit Analysis ◽

Over Time

David Lewis is widely credited with the first formulation of common knowledge and the first rigorous analysis of convention. However, common knowledge and convention entered mainstream game theory only when they were formulated, later and independently, by other theorists. As a result, some of the most distinctive and valuable features of Lewis' game theory have been overlooked. We re-examine this theory by reconstructing key parts in a more formal way, extending it, and showing how it differs from more recent game theory. In contrast to current theories of common knowledge, Lewis' theory is based on an explicit analysis of the modes of reasoning that are accessible to rational individuals and so can be used to analyse the genesis of common knowledge. Lewis' analysis of convention emphasises the role of inductive reasoning and of salience in the maintenance of conventions over time.

Download Full-text

Knowing What Others Know: Common Knowledge, Accounting, and Capital Markets

Accounting Horizons ◽

10.2308/acch.2002.16.4.305 ◽

2002 ◽

Vol 16 (4) ◽

pp. 305-318 ◽

Cited By ~ 32

Author(s):

Shyam Sunder

Keyword(s):

Game Theory ◽

Information Systems ◽

Financial Reporting ◽

Capital Markets ◽

Common Knowledge ◽

Real Life ◽

Managerial Control ◽

New Developments ◽

Business Applications ◽

Potential Applications

The concept of common knowledge concerning higher orders of knowledge has seen exciting new developments in the fields of philosophy, game theory, statistics, economics, and cognitive science in the recent decades. Even though information lies at the heart of accounting and capital markets research, these new developments have remained at the periphery of these fields. Common knowledge thinking may significantly advance our understanding of financial reporting, analysis, securities valuation, managerial control, auditing, and information systems. Such accounting and business applications will also make important contributions in the form of concrete, real-life examples and applications to the basic fields where the idea of common knowledge originated. This paper is an overview of common knowledge and its actual and potential applications to accounting and capital markets research.

Download Full-text

THE INSTABILITY OF JOHN RAWLS'S “STABILITY FOR THE RIGHT REASONS”

Episteme ◽

10.1017/epi.2017.14 ◽

2017 ◽

Vol 16 (1) ◽

pp. 1-17 ◽

Cited By ~ 4

Author(s):

Hun Chung

Keyword(s):

Game Theory ◽

Common Knowledge ◽

Political Order ◽

The Right

ABSTRACTJohn Rawls's most mature notion of political order is “stability for the right reasons.” Stability for the right reasons is the kind of political order that Rawls hoped a well-ordered society could ideally achieve. In this paper, I demonstrate through the tools of modern game theory, theinstabilityof “stability for the right reasons.” Specifically, I will show that a well-ordered society can completely destabilize by the introduction of an arbitrarily small number of non-compliers whenever individuals fail to achieve full common knowledge ever so slightly.

Download Full-text

SMALL INFINITARY EPISTEMIC LOGICS

The Review of Symbolic Logic ◽

10.1017/s1755020319000029 ◽

2019 ◽

Vol 12 (4) ◽

pp. 702-735

Author(s):

TAI-WEI HU ◽

MAMORU KANEKO ◽

NOBU-YUKI SUZUKI

Keyword(s):

Game Theory ◽

Common Knowledge ◽

Expressive Power ◽

Infinite Regress ◽

Deductive Inference ◽

Countably Infinite ◽

Negative Introspection ◽

Positive Introspection

AbstractWe develop a series of small infinitary epistemic logics to study deductive inference involving intra-/interpersonal beliefs/knowledge such as common knowledge, common beliefs, and infinite regress of beliefs. Specifically, propositional epistemic logics GL (Lα) are presented for ordinal α up to a given αo (αo ≥ ω) so that GL(L0) is finitary KDn with n agents and GL(Lα) (α ≥ 1) allows conjunctions of certain countably infinite formulae. GL(Lα) is small in that the language is countable and can be constructive. The set of formulae Lα is increasing up to α = ω but stops at ω We present Kripke-completeness for GL(Lα) for each α ≤ ω, which is proved using the Rasiowa–Sikorski lemma and Tanaka–Ono lemma. GL(Lα) has a sufficient expressive power to discuss intra-/interpersonal beliefs with infinite lengths. As applications, we discuss the explicit definability of Axioms T (truthfulness), 4 (positive introspection), 5 (negative introspection), and of common knowledge in GL(Lα) Also, we discuss the rationalizability concept in game theory in our framework. We evaluate where these discussions are done in the series GL(Lα), α ≤ ω.

Download Full-text

Rationalizability and Common Knowledge of Rationality

The Bounds of Reason ◽

10.23943/princeton/9780691160849.003.0004 ◽

2014 ◽

Author(s):

Herbert Gintis

Keyword(s):

Game Theory ◽

Normal Form ◽

Common Knowledge ◽

Analytical Framework ◽

Epistemic Game Theory ◽

Dominated Strategies ◽

Common Knowledge Of Rationality ◽

The Common ◽

Rational Players ◽

Rationality Assumption

This chapter deals with the implications of rationality in normal form games. It first explores the ramifications of the rationalizability assumption and shows that in many cases rational individuals will not play rationalizable strategies. It argues that the informal reasoning supporting rationalizability must be replaced by a more rigorous analytical framework. This framework is known as epistemic game theory. Using epistemic game theory, it presents the argument that not rationality, but rather common knowledge of rationality, implies that players will only use rationalizable strategies. The chapter concludes by showing that there is no justification of the common knowledge of rationality assumption, and hence there is no reason to believe that in general rational players will choose rationalizable strategies. It strengthens this conclusion by showing that even assuming common knowledge of rationality, there is no reason for a rational player to conform to the iterated elimination of strongly dominated strategies.

Download Full-text

Rational Ritual

10.23943/princeton/9780691158280.001.0001 ◽

2013 ◽

Cited By ~ 12

Author(s):

Michael Suk-Young Chwe

Keyword(s):

Game Theory ◽

French Revolution ◽

Cultural Practices ◽

Common Knowledge ◽

Reciprocal Relationship ◽

Super Bowl ◽

Social Sciences And Humanities ◽

The Social ◽

New Applications ◽

Ritual Speech

Why do Internet, financial service, and beer commercials dominate Super Bowl advertising? How do political ceremonies establish authority? Why does repetition characterize anthems and ritual speech? Why were circular forms favored for public festivals during the French Revolution? This book answers these questions using a single concept: common knowledge. Game theory shows that in order to coordinate its actions, a group of people must form “common knowledge.” Each person wants to participate only if others also participate. Members must have knowledge of each other, knowledge of that knowledge, knowledge of the knowledge of that knowledge, and so on. The book applies this insight to analyze a range of rituals across history and cultures. It shows that public ceremonies are powerful not simply because they transmit meaning from a central source to each audience member but because they let audience members know what other members know. For instance, people watching the Super Bowl know that many others are seeing precisely what they see and that those people know in turn that many others are also watching. This creates common knowledge, and advertisers selling products that depend on consensus are willing to pay large sums to gain access to it. Remarkably, a great variety of rituals and ceremonies, such as formal inaugurations, work in much the same way. By using a rational-choice argument to explain diverse cultural practices, the book argues for a close reciprocal relationship between the perspectives of rationality and culture. It illustrates how game theory can be applied to an unexpectedly broad spectrum of problems, while showing in an admirably clear way what game theory might hold for scholars in the social sciences and humanities who are not yet acquainted with it. A new afterword delves into new applications of common knowledge, both in the real world and in experiments, and considers how generating common knowledge has become easier in the digital age.

Download Full-text

Knowledge, Awareness and Probabilistic Beliefs

The B E Journal of Theoretical Economics ◽

10.1515/bejte-2019-0014 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Tomasz Sadzik

Keyword(s):

Game Theory ◽

Incomplete Information ◽

Common Knowledge ◽

Strategic Interaction ◽

Bayesian Game ◽

Bayesian Equilibrium ◽

Common Prior ◽

Probabilistic Beliefs ◽

Common Knowledge Of Rationality ◽

Axiom Systems

Abstract Bayesian game theory investigates strategic interaction of players with full awareness but incomplete information about their environment. We extend the analysis to players with incomplete awareness, who might not be able to reason about all contingencies in the first place. We develop three logical systems for knowledge, probabilistic beliefs and awareness, and characterize their axiom systems. Bayesian equilibrium is extended to games with incomplete awareness and we show that it is consistent with common prior and speculative trade, when common knowledge of rationality is violated.

Download Full-text