Epistemic logics and their game theoretic applications: Introduction

Summary. 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. Received: July 1, 2000; revised version: April 19, 2001     

Reasoning about knowledge of unawareness

Awareness has been shown to be a useful addition to standard epistemic logic. However, standard propositional logics for knowledge and awareness cannot express the fact that an agent knows that there are facts of which he is unaware without there being an explicit fact that the agent knows he is unaware of. We extend Fagin and Halpern's logic of general awareness to a logic that allows quantification over variables, so that there is a formula in the language that says “an agent explicitly knows that there exists a fact of which he is unaware.” Moreover, that formula can be true without the agent explicitly knowing that he is unaware of any particular formula. We provide a sound and complete axiomatization of the logic. Finally, we show that the validity problem for the logic is recursively enumerable, but not decidable.     

The probability of nontrivial common knowledge

We study the probability that two or more agents can attain common knowledge of nontrivial events when the size of the state space grows large. We adopt the standard epistemic model where the knowledge of an agent is represented by a partition of the state space. Each agent is endowed with a partition generated by a random scheme consistent with his cognitive capacity. Assuming that agentsʼ partitions are independently distributed, we prove that the asymptotic probability of nontrivial common knowledge undergoes a phase transition. Regardless of the number of agents, when their cognitive capacity is sufficiently large, the probability goes to one; and when it is small, it goes to zero. Our proofs rely on a graph-theoretic characterization of common knowledge that has independent interest.     

多重制度逻辑交互关系如何影响持续性变革过程？—基于云南农垦集团20年政企分离转型纵向案例分析

在更加动态和复杂的背景下,持续性变革取向的价值更加凸显.本文采用案例研究方法,以一家国有组织20年政企分离转型的过程为分析对象,尝试探究持续性变革的动力来源及过程机制.结合多重制度逻辑理论,研究发现小社会逻辑和市场逻辑是组织政企分离过程中最突出的两种制度逻辑.多重制度逻辑交互关系与持续性变革状态转换之间的具体关系如下:①受到外部环境影响,市场逻辑迅速提升对内部小社会逻辑产生挤压,两种制度逻辑之间的冲突显性化促使组织从"解冻"状态进入"冻结"状态,随后转向"再平衡"的变革状态;②组织内部小社会逻辑的回顾使得制度逻辑之间的冲突性逐渐减弱,兼容性逐渐凸显,推动组织由"再平衡"状态转向"解冻"状态;③多重制度逻辑交互关系的不断转换为组织持续性变革提供情景动力.研究结合组织内外部因素对持续性变革的驱动情景进行了有益探索,丰富了组织变革理论研究,同时为当前背景下企业变革和创新实践提供启示.     

A note on the wise girls puzzle

Summary. This article analyzes the two wise girls puzzle, which is a simpler variant of the so-called three wise men puzzle, with some proof-theoretic tools. We formulate the puzzle in an epistemic logic. Our chief assumption is that the reasoning ability of each player of the puzzle is equivalent to what is described by the epistemic logic. We will interpret the behaviors of the players in the puzzle in terms of unprovability of certain statements. The proof-theoretic tools we employ are consequences of a meta-theorem, known as the cut elimination theorem. Received: August 1, 2000; revised version: May 15, 2001     

The Role of Southern Intellectuals in Contemporary Trade Governance

This paper identifies the emergence of a new set of trade intellectuals from the global South, primarily from the rising powers. It argues that this group of ‘Southern trade intellectuals’ has formed a loose epistemic community, and traces the impact the group is having on global trade governance. The subject examined differs from most other uses of the epistemic community framework in that it analyses a case in which there is no objective, scientific knowledge being promoted by the group. Instead, the group is engaged in promoting one subjective understanding of events over another. However, the emergence of this epistemic community has been important in providing an alternative trade narrative that has weakened the dominance previously enjoyed by the Western powers over trade analysis. This broadening of the range of trade analysis and expert opinion forms an important, though potentially problematic, area of leadership provided by the rising powers to less developed countries.     

Belief consistency and trade consistency

Interpersonal consistency can be described in epistemic terms as a property of beliefs, or in economic terms as the impossibility of certain trades. The existence of a common prior from which all agentsʼ beliefs are derived is of the first kind. The non-existence of an agreeable bet, that is, a contingent zero-sum trade which is always favorable to all agents, is of the second kind. It is well established that these two notions of consistency are equivalent for finite type spaces but not for countable ones. We present three equivalences of epistemic consistency and economic consistency conditions for countable type spaces, defining in this way three levels of consistency of type spaces: weak consistency, consistency, and strong consistency. These three levels coincide in the finite case. We fully analyze the level of consistency of type spaces based on the knowledge structure of Rubinsteinʼs email game. The new notion of belief consistency introduced here helps to justify the requirement of boundedness of payoff functions in countable type spaces by showing that in a large class of spaces there exists an agreeable unbounded bet even when a common prior exists.     

Common belief of rationality in games of perfect information

Aumann (1995) showed that for games with perfect information common knowledge of substantive rationality implies backward induction. Substantive rationality is defined in epistemic terms, that is, in terms of knowledge. We show that when substantive rationality is defined in doxastic terms, that is, in terms of belief, then common belief of substantive rationality implies backward induction. Aumann (1998) showed that material rationality implies backward induction in the centipede game. This result does not hold when rationality is defined doxastically. However, if beliefs are interpersonally consistent then common belief of material rationality in the centipede game implies common belief of backward induction.     

Problems of Emergency Arbitration

This paper explores ethical burdens facing the economics profession which are associated with epistemic features of economic practice. Economists exert power over those they purport to serve by virtue of epistemic asymmetry between themselves and others, i.e., the intellectual monopoly they enjoy over a vitally important body of knowledge. But they also face the problem of epistemic insufficiency, which implies that they may do substantial harm as they try to do good. The paper explores the ethical entailments of the epistemic features of economics, and argues that managing the ethical challenges requires a new field of inquiry, the field of professional economic ethics, and not just a code of conduct.     

Emergence versus neoclassical reductions in economics

ABSTRACT

Many epistemic anomalies of the neoclassical research programme originate from its ontologically reductionist meta-axioms, which predicate how economic macro-systems are constituted from their micro-entities and how the latter behave – namely atomistic aggregativity, normative equilibration and global instrumental rationality. This paper explores the metaphysical foundations of the premise of emergence and argues that it can be a remedy to the ills of neoclassical reductions, and a foundational epistemic principle in a progressive systemic research programme in economics, which would bridge existing streams of ‘heterodox’ economic theory.     

Can probability theory deal with entrepreneurship?

The Austrian theory of entrepreneurship emphasizes the importance of epistemic heterogeneity and the unlistability of the set of all possibilities. A similar concern with what has been called “the art of choosing the space of possibilities” is an important part of Bayesian model selection. Both Austrian and Bayesian authors view the common knowledge assumption as an unrealistic and unnecessary restriction. This coincidence of concerns leads to a joint theory of entrepreneurship. Three important benefits result from this merger: (1) the ability to use Itti & Baldi’s Bayesian theory of surprise to empirically measure radical surprise and improve the Betrand competition model as a consequence, (2) dealing with the unlistability problem, and (3) better understanding why the emergence of common knowledge is always the outcome of a social process rather than an inherent consequence of “rationality”.     

Bounded interpersonal inferences and decision making

Summary. Individual decision making is based on predictions about other players' choices as well as on valuations of reactions to predictions. In this sense, a player has a prediction-decision criterion for decision making. We develop a theory of prediction-decision criteria, which enables us to capture new phenomena on individual decision making in games. The decision making situation is described in the epistemic logic GL of shallow depths. There, each player considers his and other players' decision making down to some shallow depths. It is a point of our theory to investigate inferential complexities of interpersonal introspections. In particular, we can discuss a minimal epistemic inferential structure for prediction-decision making. We will find parallel structures in decision making and prediction making, which is called an inner parallelism. The climax of the paper is the consideration of inner parallelisms of prediction-decision making. Received: August 31, 2000; revised version: April 9, 2001     

Consistency of beliefs and epistemic conditions for Nash and correlated equilibria

A consistency condition (action-consistency) on the interim beliefs of players in a game is introduced. Action-consistency is weaker than common priors and, unlike common priors, is characterized by a “no-bets” condition on verifiable events. Using action-consistency, we provide epistemic conditions to Nash and correlated equilibria weakening the common knowledge restrictions in Aumann and Brandenburger [Aumann, R., Brandenburger, A., 1995. Epistemic conditions for Nash equilibrium. Econometrica 63, 1161–1180] and Aumann [Aumann, R., 1987. Correlated equilibrium as an expression of Bayesian rationality. Econometrica 55, 1–18].     

Evaluations of epistemic components for resolving the muddy children puzzle

We evaluate the 3 child muddy children puzzle using the epistemic logic of shallow depths $\text{ GL}_{EF}$ . This system is used to evaluate what components are necessary for a resolution. These components include the basic beliefs of a child, the necessary depths of the epistemic structures, and the observations about the inactions of others added after a stage. These are all given explicitly, and their necessity is examined. We formulate the concept of a resolution as a process of inferences, actions, observations, and belief changes. We give three theorems. The first one gives a specific resolution, in which no common knowledge is involved. The second theorem states that any resolution has length of at least 3. The third theorem shows that the resolution given in the first theorem is minimal in various senses. In this manner, the necessary components for a resolution of the puzzle are evaluated.     

What do uncertainty-averse decision-makers believe?

Summary. This paper introduces the concept of firm belief, which is proposed as a new epistemic model for a wide class of preferences. In particular, firm beliefs are shown to have the following desirable properties: (i) they are derived from preferences according to a plausible rule of epistemic inference; (ii) they satisfy standard logical properties; and (iii) tractable representations of firm belief are available for all (suitably continuous) biseparable preferences [13, 14], including the Choquet expected utility [30] and maxmin expected utility [16] classes. We also use firm belief to construct a generalization of Nash equilibrium for (two-player) normal form games. Received: December 14, 1999; revised version: February 26, 2001     

Epistemic equivalence of extended belief hierarchies

In this paper, we introduce a notion of epistemic equivalence between hierarchies of conditional beliefs and hierarchies of lexicographic beliefs, thus extending the standard equivalence results of Halpern (2010) and Brandenburger et al. (2007) to an interactive setting, and we show that there is a Borel surjective function, mapping each conditional belief hierarchy to its epistemically equivalent lexicographic belief hierarchy. Then, using our equivalence result we construct a terminal type space model for lexicographic belief hierarchies. Finally, we show that whenever we restrict attention to full-support beliefs, epistemic equivalence between a lexicographic belief hierarchy and a conditional belief hierarchy implies that an arbitrary Borel event is commonly assumed under the lexicographic belief hierarchy if and only if it is commonly strongly believed under the conditional belief hierarchy. This is the first result in the literature directly linking common assumption in rationality (Brandenburger et al., 2008) with common strong belief in rationality (Battigalli and Siniscalchi, 2002).     

Knowledge Games

The subject of epistemic logic is firmly entrenched in game theory, including the analysis of common knowledge and of public announcements, such as in 'hat problems'. How to analyse communications to sub groups of the public, and the effects of such common knowledge of a subgroup on the information state of a larger group, has only recently come into fruition. Knowledge games are introduced to provide a comfortably concrete vehicle for the study of such interactions. This paper introduces the concepts of knowledge game, deal of cards, knowledge game state, game action, and action execution. A deal of cards is a function from cards to players. A knowledge game state is represented by a pointed multiagent S5 model on the set of card deals where all players hold the same number of cards as in the actual deal. A game action combines a question with an answer, and is represented by a pointed multiagent S5 frame on the set of possible answers. The execution of a game action in a knowledge game state corresponds to the computation of a pointed multiagent S5 model that is a restriction of the direct product of the corresponding action frame and game model.     

The influence of causation and effectuation logics on targeted policies: the cases of Singapore and Israel

We examine the causation and effectuation logics for implementing targeted biotechnology policies using two case studies: Singapore (causation) and Israel (effectuation). After more than a decade of implementing targeted biotechnology policies, both Singapore and Israel have failed to create fully fledged biotech clusters. Singapore has been unsuccessful in creating vibrant entrepreneurial activity that will support its transformation into a knowledge economy. Israel has failed to turn its 1000 small, dedicated biotechnology firms into a substantial engine of growth and employment. The paper questions how these two policy approaches influenced the targeting of the biotechnology sectors and identifies the limits of these approaches in supporting targeting. We conclude that a combination of the two logics is needed, especially when targeting complex sectors with a yet unknown development path.