scholarly journals SMALL INFINITARY EPISTEMIC LOGICS

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.

scholarly journals Game Theory Without Partitions, and Applications to Speculation and Consensus

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
John Geanakoplos
Keyword(s):  
Game Theory ◽  
Information Processing ◽  
Decision Theory ◽  
Common Knowledge ◽  
Extended Theory ◽  
Knowing That ◽  
Processing Errors ◽  
Necessary And Sufficient

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.

Game Theory and the Law

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.

Scholastická logika „vědění“ I.

2018 ◽  
Vol 15 (5) ◽  
pp. 127-205
Author(s):  
Miroslav Hanke ◽  
Keyword(s):  
Middle Ages ◽  
Epistemic Logic ◽  
Unified Framework ◽  
Founding Father ◽  
The Core ◽  
Cognitive Operations ◽  
Conceptual Background ◽  
John Wyclif ◽  
Negative Introspection ◽  
Positive Introspection

Fourteenth-century logic gave rise, among others, to the genre De scire et dubitare, which offered a unified framework for discussing different forms of epistemic sophisms by utilising the underlying systems of epistemic logic. One of the problems introduced in this context already by the founding father of this genre, William Heytesbury, was the so-called axiom of positive introspection, i.e., the principle that an agent who knows that something is the case, knows that she knows that it is the case. Owing to Heytesbury’s enormous popularity in the subsequent centuries, discussion of this problem became relatively widespread. This debate was addressed already in Boh’s seminal Epistemic Logic in the Later Middle Ages, which, despite its limitations acknowledged by its author, is a standard source. The present study elaborates on Boh by extending the corpus of his works (both in the sense of including new authors and of utilising manuscripts along with printed editions) and drawing new connections based on that. The core of the survey consists of an analysis of the positions of William Heytesbury and John Wyclif (both pertaining to the context of Merton College), their Italian reception by Peter of Mantua, and the “continental” reception of Heytesbury by John of Holland. The main goals of this study are to formalise the key arguments, which makes it possible to address the underlying systems of epistemic logic and their respective “strength”, and to articulate the conceptual background of those arguments and systems (the concepts of evidence, attention, and order of cognitive operations). The gist of the debate is, on one of the sides, an attempt to prove that it is impossible to doubt whether one knows that something is the case by employing whether the principles of positive introspection and of distribution of knowledge over implication, or the principles of positive and negative introspection combined.

Common Knowledge and Cheap Talk in Democratic Discourse and Law

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.

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

2003 ◽  
Vol 19 (2) ◽  
pp. 175-210 ◽  
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.

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

2002 ◽  
Vol 16 (4) ◽  
pp. 305-318 ◽  
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.

Is Language a Game?

1996 ◽  
Vol 26 (1) ◽  
pp. 1-28 ◽  
Author(s):  
Joseph Heath
Keyword(s):  
Game Theory ◽  
Social Life ◽  
Predicate Logic ◽  
Expressive Language ◽  
Expressive Power ◽  
Formal Models ◽  
Basic Action ◽  
Basic Assumptions ◽  
Recent Developments ◽  
Power Game

Recent developments in game theory have shown that the mathematical models of action so widely admired in the study of economics are in fact only particular instantiations of a more general theoretical framework. In the same way that Aristotelian logic was ‘translated’ into the more general and expressive language of predicate logic, the basic action theoretic underpinnings of modern economics have now been articulated within the more comprehensive language of game theory. But precisely because of its greater generality and expressive power, game theory has again revived the temptation to apply formal models of action to every domain of social life. This movement has been fuelled by some notable successes. Game theory has provided useful insights into the logic of collective action in the theory of public goods, and strategic models of voting have illustrated important aspects of institutional decision-making. But this extension of formal models into every area of social interaction has also encountered significant difficulties, despite the fact that contemporary decision theory has weakened its basic assumptions to the point where it teeters constantly on the brink of vacuity.

scholarly journals OBLIGATION, FREE CHOICE, AND THE LOGIC OF WEAKEST PERMISSIONS

2015 ◽  
Vol 8 (4) ◽  
pp. 807-827 ◽  
Author(s):  
ALBERT J.J. ANGLBERGER ◽  
NOBERT GRATZL ◽  
OLIVIER ROY
Keyword(s):  
Game Theory ◽  
Free Choice ◽  
Expressive Power ◽  
Axiom System ◽  
Deontic Modals ◽  
Neighborhood Models

AbstractWe introduce a new understanding of deontic modals that we callobligations as weakest permissions. We argue for its philosophical plausibility, study its expressive power in neighborhood models, provide a complete Hilbert-style axiom system for it and show that it can be extended and applied to practical norms in decision and game theory.

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

Episteme ◽  
2017 ◽  
Vol 16 (1) ◽  
pp. 1-17 ◽  
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.

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