### Every hierarchy of beliefs is type ∗

, 2008

"... Any model of incomplete information situations has to consider the players ’ hierarchies of beliefs, which can make the modeling very cumbersome. Harsányi [12] suggested that the hierarchies of beliefs can be replaced by types, i.e., a type space can substitute for the hierarchies of beliefs (hencef ..."

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Any model of incomplete information situations has to consider the players ’ hierarchies of beliefs, which can make the modeling very cumbersome. Harsányi [12] suggested that the hierarchies of beliefs can be replaced by types, i.e., a type space can substitute for the hierarchies of beliefs (henceforth Harsányi program). In the purely measurable framework Heifetz and Samet [15] formalized the concept of type space, and proved that there is universal type space, i.e., the most general type space exists. Later Meier [17] showed that the universal type space is complete, in other words, the universal type space is a consistent object. After these results, only one step is missing to prove that the Harsányi program works, that every hierarchy of beliefs is in the complete universal type space, put it differently, every hierarchy of beliefs can be replaced by type. In this paper we also work in the purely measurable framework, and show that the types can substitute for all hierarchies of beliefs, i.e., the Harsányi program works. 1

### and

, 2005

"... In this paper we explore the mechanisms that allow securities analysts to value companies in contexts of Knightian uncertainty, that is, in the face of information that is unclear, subject to unforeseeable contingenc ies or to multiple interpretations. We address this question with a grounded-theory ..."

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In this paper we explore the mechanisms that allow securities analysts to value companies in contexts of Knightian uncertainty, that is, in the face of information that is unclear, subject to unforeseeable contingenc ies or to multiple interpretations. We address this question with a grounded-theory analysis of the reports written on Amazon.com by securities analyst Henry Blodget and rival analysts during the years 1998-2000. Our core finding is that analysts ’ reports are structured by internally consistent associations that include categorizations, key metrics and analogies. We refer to these representations as calculative frames, and propose that analysts function as frame-makers – that is, as specialized intermediaries that help investors value uncertain stocks. We conclude by considering the implications of frame-making for the rise of new industry categories, analysts ’ accuracy, and the regulatory debate on analysts’ independence. 2 Whether as brokers, critics or analysts, market intermediaries have become increasingly

### The Language of Game Theory: Putting Epistemics into the Mathematics of Games ∗

"... Dedicated to my daughter ..."

### without counterfactuals

"... dynamic epistemic characterization of backward induction ..."

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### Interactive Rationality and the Dynamics of Reasons Interactive Rationality and the Dynamics of Reasons]Interactive Rationality and the Dynamics of Reasons Roy, Pacuit]Olivier Roy

"... What does rationality require of an individual decision maker when the consequences of her choices depend on what other rational decision makers decide? Are these requirements different from those that arise in general situations of decision making under risk or uncertainty? This paper argues that t ..."

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What does rationality require of an individual decision maker when the consequences of her choices depend on what other rational decision makers decide? Are these requirements different from those that arise in general situations of decision making under risk or uncertainty? This paper argues that they are. The argument is cast in contemporary epistemic game theory. We first explain how choice rules and solution concepts can be seen as potential sources of rational recommendations, singling out what counts as reasons for action in interactive situations. We then analyze responsiveness to such reasons, from a dynamic perspective. We argue that, against this background, basic observations and theorems from epistemic game theory turn out to be normatively significant. To illustrate that we show how to connect “objective ” reasons with what we call ex post assessments, and how to extract rational requirements of responsiveness

### Epistemic Logic and the Foundations of Decision and Game Theory

, 2010

"... This paper reviews a number of foundational results at the intersection of epistemic logic, decision and game theory. It first presents a decisiontheoretic underpinning of epistemic analysis in terms of modal operators. Then it moves to the theory of strategic interaction, and show how this kind of ..."

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This paper reviews a number of foundational results at the intersection of epistemic logic, decision and game theory. It first presents a decisiontheoretic underpinning of epistemic analysis in terms of modal operators. Then it moves to the theory of strategic interaction, and show how this kind of epistemic analysis sheds light on the assumptions underlying wellknown solution concepts, both for normal and extensive form games. Foundational work in decision and game theory aims at making explicit the assumptions that underlie the basic concepts in these disciplines. Decision theory and game theory are two sub-fields of micro-economics, concerned with the question of rational decision making by individual agents. Decision theory is the study of individual decision making under uncertainty, while game theory looks at situations of strategic interaction. Epistemic assumptions occupy a central place in the foundations of both decision theory and game theory. “Epistemic” is here understood in a broad sense, as encompassing a whole array of informational

### Epistemic Foundations of Game Theory

"... Foundational work in game theory aims at making explicit the assumptions that underlie the basic concepts of the discipline. Non-cooperative game theory is the study of individual, rational decision making in situations of strategic interaction. This entry presents the epistemic foundations of non-c ..."

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Foundational work in game theory aims at making explicit the assumptions that underlie the basic concepts of the discipline. Non-cooperative game theory is the study of individual, rational decision making in situations of strategic interaction. This entry presents the epistemic foundations of non-cooperative game theory (this area of research is called epistemic game theory). Epistemic game theory views rational decision making in games as something not essentially different from rational decision making under uncertainty. As in Decision Theory (Peterson, 2009), to choose rationally in a game is to select the “best ” action in light of one’s beliefs or information. In a decision problem, the decision maker’s beliefs are about a passive state of nature, the state of which determines the consequences of her actions. In a game, the consequences of one’s decision depend on the choices of the other agents involved in the situation (and possibly the state of nature). Recognizing this—i.e., that one is interacting with other agents who try to choose the best course of action in the light of their own beliefs—brings higher-order information into the picture. The players ’ beliefs

### LMU Munich

"... Abstract. This paper studies the following interpretation of obligations: A person i ought to do A in a situation S just in case everything else i may (and can) do in S is consistent with A. In such case A can be called the weakest permission that i has in S. We show that, under this interpretation, ..."

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Abstract. This paper studies the following interpretation of obligations: A person i ought to do A in a situation S just in case everything else i may (and can) do in S is consistent with A. In such case A can be called the weakest permission that i has in S. We show that, under this interpretation, obligation and permission are not dual notions, and that it gives rise to an interesting interplay between deontic and alethic notions. We also discuss the logics ’ adequacy w.r.t. the paradoxes of (classic) deontic logic and provide a sound and complete axiomatization for it. We finally show that practical, rational recommendations in games provide a natural, concrete application of such understanding of obligations and permissions. 1 Introduction and

### Abstract

"... We consider continuous state, continuous action batch reinforcement learning where the goal is to learn a good policy from a sufficiently rich trajectory generated by some policy. We study a variant of fitted Q-iteration, where the greedy action selection is replaced by searching for a policy in a r ..."

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We consider continuous state, continuous action batch reinforcement learning where the goal is to learn a good policy from a sufficiently rich trajectory generated by some policy. We study a variant of fitted Q-iteration, where the greedy action selection is replaced by searching for a policy in a restricted set of candidate policies by maximizing the average action values. We provide a rigorous analysis of this algorithm, proving what we believe is the first finite-time bound for value-function based algorithms for continuous state and action problems. 1

### Understanding the Brandenburger-Keisler Belief Paradox

, 2006

"... staff.science.uva.nl/∼epacuit ..."

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