In this paper, we investigate the semantics and logic of choice-driven counterfactuals, that is, of counterfactuals whose evaluation relies on auxiliary premises about how agents are expected to act, i.e., about their default choice behavior. To do this, we merge one of the most prominent logics of agency in the philosophical literature, namely stit logic, with the well-known logic of counterfactuals due to Stalnaker and Lewis. A key component of our semantics for counterfactuals is to distinguish between deviant and non-deviant (...) actions at a moment, where an action available to an agent at a moment is deviant when its performance does not agree with the agent’s default choice behavior at that moment. After developing and axiomatizing a stit logic with action types, instants, and deviant actions, we study the philosophical implications and logical properties of two candidate semantics for choice-driven counterfactuals, one called rewind models inspired by Lewis, 455–476 1979) and the other called independence models motivated by well-known counterexamples to Lewis’s proposal Slote, 3–27 1978). In the last part of the paper we consider how to evaluate choice-driven counterfactuals at moments arrived at by some agents performing a deviant action. (shrink)

We formalise a notion of dynamic rationality in terms of a logic of conditional beliefs on (doxastic) plausibility models. Similarly to other epistemic statements (e.g. negations of Moore sentences and of Muddy Children announcements), dynamic rationality changes its meaning after every act of learning, and it may become true after players learn it is false. Applying this to extensive games, we “simulate” the play of a game as a succession of dynamic updates of the original plausibility model: the epistemic situation (...) when a given node is reached can be thought of as the result of a joint act of learning (via public announcements) that the node is reached. We then use the notion of “stable belief”, i.e. belief that is preserved during the play of the game, in order to give an epistemic condition for backward induction: rationality and common knowledge of stable belief in rationality. This condition is weaker than Aumann’s and compatible with the implicit assumptions (the “epistemic openness of the future”) underlying Stalnaker’s criticism of Aumann’s proof. The “dynamic” nature of our concept of rationality explains why our condition avoids the apparent circularity of the “backward induction paradox”: it is consistent to (continue to) believe in a player’s rationality after updating with his irrationality. (shrink)

We formalise a notion of dynamic rationality in terms of a logic of conditional beliefs on plausibility models. Similarly to other epistemic statements, dynamic rationality changes its meaning after every act of learning, and it may become true after players learn it is false. Applying this to extensive games, we "simulate" the play of a game as a succession of dynamic updates of the original plausibility model: the epistemic situation when a given node is reached can be thought of as (...) the result of a joint act of learning that the node is reached. We then use the notion of "stable belief, i.e. belief that is preserved during the play of the game, in order to give an epistemic condition for backward induction: rationality and common knowledge of stable belief in rationality. This condition is weaker than Aumann's and compatible with the implicit assumptions underlying Stalnaker's criticism of Aumann's proof. The "dynamic" nature of our concept of rationality explains why our condition avoids the apparent circularity of the "backward induction paradox": it is consistent to believe in a player's rationality after updating with his irrationality. (shrink)

In previous work I’ve defended an interest-relative theory of belief. This paper continues the defence. It has four aims. -/- 1. To offer a new kind of reason for being unsatis ed with the simple Lockean reduction of belief to credence. 2. To defend the legitimacy of appealing to credences in a theory of belief. 3. To illustrate the importance of theoretical, as well as practical, interests in an interest-relative account of belief. 4. To revise my account to cover propositions (...) that are practically and theoretically irrelevant to the agent. (shrink)

Michael Strevens’s book Depth is a great achievement.1 To say anything interesting, useful and true about explanation requires taking on fundamental issues in the metaphysics and epistemology of science. So this book not only tells us a lot about scientific explanation, it has a lot to say about causation, lawhood, probability and the relation between the physical and the special sciences. It should be read by anyone interested in any of those questions, which includes presumably the vast majority of readers (...) of this journal. One of its many virtues is that it lets us see more clearly what questions about explanation, causation, lawhood and so on need answering, and frames those questions in perspicuous ways. I’m going to focus on one of these questions, what I’ll call the Goldilocks problem. As it turns out, I’m not going to agree with all the details of Strevens’s answer to this problem, though I suspect that something like his answer is right. At least, I hope something like his answer is right; if it isn’t, I’m not sure where else we can look. (shrink)

Two justifications of backward induction (BI) in generic perfect information games are formulated using Bonanno's (1992; Theory and Decision 33, 153) belief systems. The first justification concerns the BI strategy profile and is based on selecting a set of rational belief systems from which players have to choose their belief functions. The second justification concerns the BI path of play and is based on a sequential deletion of nodes that are inconsistent with the choice of rational belief functions.

Adam Brandenburger and H. Jerome Keisler have recently discovered a two person Russell-style paradox. They show that the following configurations of beliefs is impossible: Ann believes that Bob assumes that Ann believes that Bob’s assumption is wrong. In [7] a modal logic interpretation of this paradox is proposed. The idea is to introduce two modal operators intended to represent the agents’ beliefs and assumptions. The goal of this paper is to take this analysis further and study this paradox from the (...) point of view of a modal logician. In particular, we show that the paradox can be seen as a theorem of an appropriate hybrid logic. (shrink)

This paper proposes a logical framework for representing static and dynamic properties of different kinds of individual and collective attitudes. A complete axiomatization as well as a decidability result for the logic are given. The logic is applied to game theory by providing a formal analysis of the epistemic conditions of iterated deletion of weakly dominated strategies (IDWDS), or iterated weak dominance for short. The main difference between the analysis of the epistemic conditions of iterated weak dominance given in this (...) paper and other analysis is that we use a semi-qualitative approach to uncertainty based on the notion of plausibility first introduced by Spohn, whereas other analysis are based on a quantitative representation of uncertainty in terms of probabilities. (shrink)

We propose a minimal logic for interactive epistemology based on a qualitative representation of epistemic individual and group attitudes including knowledge, belief, strong belief, common knowledge and common belief. We show that our logic is sufficiently expressive to provide an epistemic foundation for various game-theoretic solution concepts including “1-round of deletion of weakly dominated strategies, followed by iterated deletion of strongly dominated strategies” ) and “2-rounds of deletion of weakly dominated strategies, followed by iterated deletion of strongly dominated strategies” ). (...) Axiomatization and complexity results for the logic are given in the paper. (shrink)

One of the main contributions of Richard Bradley’s book is an elegant extension of Jeffrey’s Logic of Decision that countenances the evaluation of conditional prospects. This extension offers a promising new setting in which to model dynamic choice. In Bradley’s framework, plans can be understood as conditionals of an appropriate sort, while dynamic consistency can be viewed as providing a constraint on the evaluation of conditionals across time. In this paper, we study connections between planning conditionals and dynamic consistency.

Two philosophical questions arise about rationality in centipede games that are logically prior to attempts to apply the formal tools of game theory to this topic. First, given that the players have common knowledge of mutual rationality and common knowledge that they are each motivated solely to maximize their own profits, is there a backwards-induction argument that employs only familiar non-technical concepts about rationality, leads to the conclusion that the first player is rationally obligated to end the game at the (...) first step, is deductively valid, employs premises all of which are prima facie highly plausible, and is prima facie sound and )? Second, if there is such an argument, then is it actually sound, or is it instead defective somehow despite being prima facie sound? Addressing these two questions is our project. We present a backwards-induction argument that is prima facie sound; we argue that it is an instance of the notorious sorites paradox, and hence that the concepts of rational obligatoriness and rational permissibility are vague; and we briefly address the potential consequences of all this for the foundations of game theory and decision theory. (shrink)

This article proposes two interpretations of the concept of belief hierarchies in Bayesian game theory: the behaviorist interpretation and the mentalist interpretation. On the former, belief hierarchies are derived from the players’ preferences over acts. On the latter, they are causal mechanisms that are responsible for the players’ choices and preferences over acts. The claim is that the epistemic program in game theory is potentially confronted with a dilemma regarding which interpretation should be adopted. If the behaviorist interpretation of belief (...) hierarchies is retained, then Bayesian game theory cannot account for the players’ reasoning that leads to the generation of specific belief hierarchies and thus to the implementation of a specific strategy profile in a game. If the mentalist interpretation is retained, then belief hierarchies appear to be unnecessarily and unrealistically complex objects and that as a result they fail to provide reasonable epistemic foundations for various solution concepts. In other words, neither interpretation seems to allow the epistemic program to fulfill all its aims. I suggest however that the mentalist interpretation can be arranged to escape this dilemma. (shrink)

This article proposes two interpretations of the concept of belief hierarchies in Bayesian game theory: the behaviorist interpretation and the mentalist interpretation. On the former, belief hierarchies are derived from the players’ preferences over acts. On the latter, they are causal mechanisms that are responsible for the players’ choices and preferences over acts. The claim is that the epistemic program in game theory is potentially confronted with a dilemma regarding which interpretation should be adopted. If the behaviorist interpretation of belief (...) hierarchies is retained, then Bayesian game theory cannot account for the players’ reasoning that leads to the generation of specific belief hierarchies and thus to the implementation of a specific strategy profile in a game. If the mentalist interpretation is retained, then belief hierarchies appear to be unnecessarily and unrealistically complex objects and that as a result they fail to provide reasonable epistemic foundations for various solution concepts. In other words, neither interpretation seems to allow the epistemic program to fulfill all its aims. I suggest however that the mentalist interpretation can be arranged to escape this dilemma. (shrink)

The Standard View is that, other things equal, speakers’ judgments about the meanings of sentences of their language are correct. After all, we make the meanings, so how wrong can we be about them? The Standard View underlies the Elicitation Method, a typical method in semantic fieldwork, according to which we should work out the truth-conditions of a sentence by eliciting speakers’ judgments about its truth-value in different situations. I put pressure on the Standard View and therefore on the Elicitation (...) Method: for quite straightforward reasons, speakers can be radically mistaken about meanings. -/- Lewis gave a theory of convention in a game-theoretic framework. He showed how conventions could arise from repeated coordination games, and, as a special case, how meanings could arise from repeated signaling games. I put pressure on the Standard View by building on Lewis’s framework. I construct coordination games in which the players can be wrong about their conventions, and signaling games in which the players can be wrong about their messages’ meanings. The key idea is straightforward: knowing your own strategy and payoff needn’t determine what the others do, so leaves room for false beliefs about the convention and meanings. The examples are simple, explicit, new in kind, and based on an independently plausible meta-semantic story. (shrink)

Contents. Introduction. 1. Preliminaries. 2. Normal Form Games. 3. Extensive Games. 4. Applications of Game Theory. 5. The Methodology of Game Theory. Conclusion. Appendix. Bibliography. Index. Does game theory—the mathematical theory of strategic interaction—provide genuine explanations of human behaviour? Can game theory be used in economic consultancy or other normative contexts? Explaining Games: The Epistemic Programme in Game Theory—the first monograph on the philosophy of game theory—is an attempt to combine insights from epistemic logic and the philosophy of science to (...) investigate the applicability of game theory in such fields as economics, philosophy and strategic consultancy. I prove new mathematical theorems about the beliefs, desires and rationality principles of individual human beings, and explore in detail the logical form of game theory as it is used in explanatory and normative contexts. I argue that game theory reduces to rational choice theory if used as an explanatory device, and that game theory is nonsensical if used as a normative device. A provocative account of the history of game theory reveals that this is not bad news for all of game theory, though. Two central research programmes in game theory tried to find the ultimate characterisation of strategic interaction between rational agents. Yet, while the Nash Equilibrium Refinement Programme has done badly thanks to such research habits as overmathematisation, model-tinkering and introversion, the Epistemic Programme, I argue, has been rather successful in achieving this aim. "The 'epistemic' approach to game theory has emerged over the past twenty-five years. What is this approach? How does it differ from the conventional equilibrium-based approach to game theory? What have been its strengths and weaknesses to date? To find out, read this comprehensive and excellently written account". Adam Brandenburger, J. P. Valles Professor of Business Economics and Strategy, Stern School of Business, New York University "Reading Boudewijn de Bruin's book should be rewarding both for game theorists interested in the conceptual foundations of their discipline and for philosophers who want to learn more about formal analysis of strategic interaction. It provides an in-depth logical study of the currently dominant epistemic approaches to non-cooperative games, with an eye both to the attractions and to the serious challenges facing the Epistemic Programme". Wlodek Rabinowicz, Professor of Practical Philosophy, Department of Philosophy, Lund University . (shrink)

This volume presents mathematical game theory as an interface between logic and philosophy.

By Belief Revision it is understood a system that logically explains the rational process of changing beliefs by taking into account a new piece of information. The most influential approach in this field of study, the AGM system, proposed by Alchourrón, Gärdenfors, and Makinson, postulates rationality criteria for different types of belief change. In this paper I shall assess the relationship between those criteria and argue for an opposition between the principles of Information Economy and Consistency. Furthermore, I shall argue (...) that Paraconsistent Belief Revision manages to minimise this friction in the best possible way. (shrink)

We analyze the sequential structure of dynamic games with perfect information. A three-stage account is proposed, that species setup, reasoning and play stages. Accordingly, we define a player as a set of agents corresponding to these three stages. The notion of agent connectedness is introduced into a type-based epistemic model. Agent connectedness measures the extent to which agents' choices are sequentially stable. Thus describing dynamic games allows to more fully understand strategic interaction over time. In particular, we provide suffcient conditions (...) for backward induction in terms of agent connectedness. Also, our framework makes explicit that the epistemic independence assumption involved in backward induction reasoning is stronger than usually presumed, and makes accessible multiple-self interpretations for dynamic games. (shrink)