There are two general classes of social conventions: conventions of coordination, and conventions of partial conflict. In coordination problems, the interests of the agents coincide, while in partial conflict problems, some agents stand to gain only if other agents unilaterally make certain sacrifices. Lewis' (1969) pathbreaking analysis of convention in terms of game theory focuses on coordination problems, and cannot accommodate partial conflict problems. In this paper, I propose a new game-theoretic definition of convention which generalizes previous game-theoretic definitions (Lewis (...) 1969, Vanderschraaf 1995), and which can be used to characterize norms of justice in partial conflict situations. I argue that the key structural property necessary for a social arrangement to be a convention is that it be conditionally self-enforcing, in the sense that: (i) each agent has a decisive reason to follow her end of the arrangement given that she expects all to do likewise, (ii) given a different set of expectations, some agents would have had a decisive reason to deviate, and (iii) these facts are common knowledge. This leads to a definition of convention as a strict correlated equilibrium (Aumann 1974) together with appropriate common knowledge conditions. Examples are given in which it is shown how this more general account of convention can be used to analyze norms of justice as well as coordination problems. It is only a general sense of common interest; which sense all the members of the society express to one another, and which induces them to regulate their conduct by certain rules. I observe, that it will be for my interest to leave another in the possession of his goods, provided he will act in the same manner with regard to me. He is sensible of a like interest in the regulation of his conduct. When this common sense of interest is mutually express'd, and is known to both, it produces a suitable resolution and behavior. And this may properly be call'd a convention or agreement betwixt us,.. (shrink)

For as long as there have been theories about common knowledge, they have been exposed to a certain amount of skepticism. Recent more sophisticated arguments question whether agents can acquire common attitudes and whether they are needed in social reasoning. I argue that this skepticism arises from assumptions about practical reasoning that, considered in themselves, are at worst implausible and at best controversial. A proper approach to the acquisition of attitudes and their deployment in decision making leaves room for common (...) attitudes. Postulating them is no worse off than similar idealizations that are usefully made in logic and economics. (shrink)

Famously, Kripke has argued that the central portion of the Philosophical Investigations describes both a skeptical paradox and its skeptical solution. Solving the paradox involves the element of the community, which determines correctness conditions for rule-following behavior. What do such conditions precisely consist of? Is it accurate to say that there is no fact to the matter of rule following? How are the correctness conditions sustained in the community? My answers to these questions revolve around the idea that a rule (...) is followed insofar as a convention is in place. In particular, I consider the game-theoretic definition of convention offered by David Lewis and I show that it illuminates essential aspects of the communitarian understanding of rule-following. Make the following experiment: say “It’s cold here” and mean “It’s warm here”. Can you do it?Ludwig Wittgenstein, Philosophical Investigations, 1953, §510.I can’t say “it’s cold here” and mean “it’s warm here”—at least, not without a little help from my friends.David Lewis, Convention. (shrink)

This paper investigates the epistemic assumptions that David Lewis makes in his account of social conventions. In particular, I focus on the assumption that the agents have common knowledge of the convention to which they are parties. While evolutionary analyses show that the common knowledge assumption is unnecessary in certain classes of games, Lewis’ original account (and, more recently, Cubitt and Sugden’s reconstruction) stresses the importance of including it in the definition of convention. I discuss arguments pro et contra to (...) argue that, although the assumption might be relevant to a descriptively adequate account of social convention, it is not required for its rational reconstruction. I then point out that Lewis’ account, properly speaking, is of common reason to believe, rather than of common knowledge, and argue that, in order to formalize aptly the distinction between reason to believe and belief, standard formal epistemic models need to be supplemented with so-called awareness structures. Finally, I stress that the notion of knowledge implicit in Lewis’ text involves interesting elements that cannot be captured in the standard propositional formalizations, but need the full expressive force of quantified epistemic logic. (shrink)

Ordinary common knowledge is formally expressed by strong probabilistic common belief. How strong exactly? The question can be answered by drawing from the similar equivalence, recently explored, between plain and probabilistic individual beliefs. I argue that such a move entails that common knowledge displays a double fragility: as a description of a collective state and as a phenomenon, because it can respectively disappear as group size increases, or more worryingly as the epistemic context changes. I argue that despite this latter (...) fragility, the effects of common knowledge on action are robust. Unfortunately, this in turn leads to a third fragility, that of the concept of common knowledge, which threatens to collapse on probabilistic common belief. This also reveals a disanalogy between the individual and the collective cases. I finally pinpoint the subtle difference entailed by the two concepts, expressed in terms of the attitude towards contrary evidence or of the agents’ awareness. As a result, common knowledge can be defended as a concept, which refers to a fragile yet distinct collective attitude. (shrink)

Defined and formalized several decades ago, widely used in philosophy and game theory, the concept of common knowledge is still considered as problematic, although not always for the right reasons. I suggest that the epistemic status of a group of human agents in a state of common knowledge has not been thoroughly analyzed. In particular, every existing account of common knowledge, whether formal or not, is either too strong to fit cognitively limited individuals, or too weak to adequately describe their (...) state. I provide a realistic definition of common knowledge, based on a formalization of David Lewis’ seminal account and show that it is formally equivalent to probabilistic common belief. This leads to a philosophical analysis of common knowledge which answers several common criticisms and sheds light on its nature. (shrink)

This essay develops a joint theory of rational (all-or-nothing) belief and degrees of belief. The theory is based on three assumptions: the logical closure of rational belief; the axioms of probability for rational degrees of belief; and the so-called Lockean thesis, in which the concepts of rational belief and rational degree of belief figure simultaneously. In spite of what is commonly believed, this essay will show that this combination of principles is satisfiable (and indeed nontrivially so) and that the principles (...) are jointly satisfied if and only if rational belief is equivalent to the assignment of a stably high rational degree of belief. Although the logical closure of belief and the Lockean thesis are attractive postulates in themselves, initially this may seem like a formal “curiosity”; however, as will be argued in the rest of the essay, a very reasonable theory of rational belief can be built around these principles that is not ad hoc and that has various philosophical features that are plausible independently. In particular, this essay shows that the theory allows for a solution to the Lottery Paradox, and it has nice applications to formal epistemology. The price that is to be paid for this theory is a strong dependency of belief on the context, where a context involves both the agent's degree of belief function and the partitioning or individuation of the underlying possibilities. But as this essay argues, that price seems to be affordable. (shrink)

Some people commonly know a proposition just in case they all know it, they all know that they all know it, they all know that they all know that they all know it, and so on. They commonly believe a proposition just in case they all believe it, they all believe that they all believe it, they all believe that they all believe that they all believe it, and so on. A long tradition in economic theory, theoretical computer science, linguistics (...) and philosophy has held that people have some approximation of common knowledge or common belief in a range of circumstances, for example, when they are looking at an object together, or when they have just discussed something explicitly in conversation. In this paper, I argue that people do not have any approximation of common knowledge or common belief in these circumstances. The argument suggests that people never have any approximation of common knowledge or common belief. (shrink)

The coordinated attack scenario and the electronic mail game are two paradoxes of common knowledge. In simple mathematical models of these scenarios, the agents represented by the models can coordinate only if they have common knowledge that they will. As a result, the models predict that the agents will not coordinate in situations where it would be rational to coordinate. I argue that we should resolve this conflict between the models and facts about what it would be rational to do (...) by rejecting common knowledge assumptions implicit in the models. I focus on the assumption that the agents have common knowledge that they are rational, and provide models to show that denying this assumption suffices for a resolution of the paradoxes. I describe how my resolution of the paradoxes fits into a general story about the relationship between rationality in situations involving a single agent and rationality in situations involving many agents. (shrink)

The two of us commonly know a proposition just in case (i) we both know it, (ii) we both know that we both know it, (iii) we both know that we both know that we both know it, and so on. In a recent paper titled ‘Uncommon Knowledge’, Harvey Lederman (2018) argues against the possibility of common knowledge. His argument rests on the empirical claim that there are minor individual variations in how we perceive things. This motivates a principle about (...) perception: when two people are perceiving something and it perceptually appears a certain way to one of them, then for all that person knows, it perceptually appears a slightly different way to the other. In this paper, I challenge Lederman’s perceptual principle and thereby his argument. In particular, I argue that there are some circumstances in which things perceptually appear a certain way to me and nonetheless I know that they don’t perceptually appear in a slightly different way to you. Indeed, I argue that not only are there exceptions to the perceptual principle, but they are widespread. (shrink)

We define infinitary extensions to classical epistemic logic systems, and add also a common belief modality, axiomatized in a finitary, fixed-point manner. In the infinitary K system, common belief turns to be provably equivalent to the conjunction of all the finite levels of mutual belief. In contrast, in the infinitary monotonic system, common belief implies every transfinite level of mutual belief but is never implied by it. We conclude that the fixed-point notion of common belief is more powerful than the (...) iterative notion of common belief. (shrink)

This paper introduces Agreement Theorems to dynamic-epistemic logic. We show first that common belief of posteriors is sufficient for agreement in epistemic-plausibility models, under common and well-founded priors. We do not restrict ourselves to the finite case, showing that in countable structures the results hold if and only if the underlying plausibility ordering is well-founded. We then show that neither well-foundedness nor common priors are expressible in the language commonly used to describe and reason about epistemic-plausibility models. The static agreement (...) result is, however, finitely derivable in an extended modal logic. We provide the full derivation. We finally consider dynamic agreement results. We show they have a counterpart in epistemic-plausibility models, and provide a new form of agreements via public announcements. (shrink)

Gupta has proposed a definition of strategic rationality cast in the framework of his revision theory of truth. His analysis, relative to a class of normal form games in which all players have a strict best reply to all other players’ strategy profiles, shows that game-theoretic concepts have revision-theoretic counterparts. We extend Gupta’s approach to deal with normal form games in which players’ may have weak best replies. We do so by adapting intuitions relative to Nash equilibrium refinements to the (...) revision-theoretic framework. We prove that there is a precise equivalence between trembling-hand perfect equilibria in two-player normal games and a revision-theoretic property. We then introduce lexicographic choice of action as a way to represent players’ expectations, which allows our analysis to reach full generality. Finally, we provide an example of the versatility of revision theory as applied to strategic interaction by formalizing a risk-and-compensation procedure of strategic choice in the revision-theoretic framework. (shrink)

This paper supersedes an ealier version, entitled "A Non-Standard Semantics for Inexact Knowledge with Introspection", which appeared in the Proceedings of "Rationality and Knowledge". The definition of token semantics, in particular, has been modified, both for the single- and the multi-agent case.

Game theory has proved a useful tool in the study of simple economic models. However, numerous foundational issues remain unresolved. The situation is particularly confusing in respect of the non-cooperative analysis of games with some dynamic structure in which the choice of one move or another during the play of the game may convey valuable information to the other players. Without pausing for breath, it is easy to name at least 10 rival equilibrium notions for which a serious case can (...) be made that here is the “right” solution concept for such games. (shrink)

Do conventions need to be common knowledge in order to work? David Lewis builds this requirement into his definition of a convention. This paper explores the extent to which his approach finds support in the game theory literature. The knowledge formalism developed by Robert Aumann and others militates against Lewis’s approach, because it shows that it is almost impossible for something to become common knowledge in a large society. On the other hand, Ariel Rubinstein’s Email Game suggests that coordinated action (...) is no less hard for rational players without a common knowledge requirement. But an unnecessary simplifying assumption in the Email Game turns out to be doing all the work, and the current paper concludes that common knowledge is better excluded from a definition of the conventions that we use to regulate our daily lives. (shrink)

According to decision theory, the rational initial action in a sequential decision-problem may be found by backward induction or folding back. But the reasoning which underwrites this claim appeals to the agent's beliefs about what she will later believe, about what she will later believe she will still later believe, and so forth. There are limits to the depth of people's beliefs. Do these limits pose a threat to the standard theory of rational sequential choice? It is argued, first, that (...) the traditional solutions of certain games depend on knowledge which exceeds depth limits, and that these solutions therefore cannot be shown rational in the usual sense. Then, for that related reason even folding back solutions of one-person problems cannot be! A revision of our notion of rational choice is proposed, analogous to the reliabilist account of knowledge of Goldman and others, by which this paradox is resolved. (shrink)

The notions of common knowledge or common belief play an important role in several areas of computer science , in philosophy, game theory, artificial intelligence, psychology and many other fields which deal with the interaction within a group of “agents”, agreement or coordinated actions. In the following we will present several deductive systems for common knowledge above epistemic logics –such as K, T, S4 and S5 –with a fixed number of agents. We focus on structural and proof-theoretic properties of these (...) calculi. (shrink)

In The Grammar of Society, first published in 2006, Cristina Bicchieri examines social norms, such as fairness, cooperation, and reciprocity, in an effort to understand their nature and dynamics, the expectations that they generate, and how they evolve and change. Drawing on several intellectual traditions and methods, including those of social psychology, experimental economics and evolutionary game theory, Bicchieri provides an integrated account of how social norms emerge, why and when we follow them, and the situations where we are most (...) likely to focus on relevant norms. Examining the existence and survival of inefficient norms, she demonstrates how norms evolve in ways that depend upon the psychological dispositions of the individual and how such dispositions may impair social efficiency. By contrast, she also shows how certain psychological propensities may naturally lead individuals to evolve fairness norms that closely resemble those we follow in most modern societies. (shrink)

Brian Skyrms constructs a theory of "dynamic deliberation" and uses it to investigate rational decision-making in cases of strategic interaction. This illuminating book will be of great interest to all those in many disciplines who use decision theory and game theory to study human behavior and thought. Skyrms begins by discussing the Bayesian theory of individual rational decision and the classical theory of games, which at first glance seem antithetical in the criteria used for determining action. In his effort to (...) show how methods for dealing with information feedback can be productively combined, the author skillfully leads us through the mazes of equilibrium selection, the Nash equilibria for normal and extensive forms, structural stability, causal decision theory, dynamic probability, the revision of beliefs, and, finally, good habits for decision. The author provides many clarifying illustrations and a handy appendix called "Deliberational Dynamics on Your Personal Computer." His powerful model has important implications for understanding the rational origins of convention and the social contract, the logic of nuclear deterrence, the theory of good habits, and the varied strategies of political and economic behavior. (shrink)

Analyzes the nature of international disagreements and conflict resolution in terms of game theory and non-zero-sum games.

It is advisable to treat some sorts of discourse about fiction with the aid of an intensional operator "in such-And-Such fiction...." the operator may appear either explicitly or tacitly. It may be analyzed in terms of similarity of worlds, As follows: "in the fiction f, A" means that a is true in those of the worlds where f is told as known fact rather than fiction that differ least from our world, Or from the belief worlds of the community in (...) which the fiction originated. (shrink)

In this dissertation, I develop a theory of rational inductive deliberation in the context of strategic interaction that generalizes previous theories of inductive deliberation. In this account of inductive deliberation, I model rational deliberators as players engaged in noncooperative games, such that: They are Bayesian rational, in the sense that every deliberator chooses actions that maximize expected utility given the beliefs this deliberator has regarding the other deliberators, and They update their beliefs about one another recursively, using rules of inductive (...) logic. Inductive deliberators update their beliefs until they reach an equilibrium of the game, at which every deliberator maximizes expected utility given the deliberator's beliefs over the actions of the other deliberators. ;The theory of inductive deliberation generalizes previous theories by allowing for the possibility of correlation in the beliefs of the deliberators. Most of the results of noncooperative game theory presuppose that the players strategies are probabilistically independent. I argue that such probabilistic independence assumptions are unfounded, and that agents should take into account the possibility that their opponents' actions are correlated. Relaxing the probabilistic independence assumption in noncooperative game theory leads to various correlated equilibrium concepts, which I argue are the appropriate solution concepts for noncooperative games. Relaxing the independence assumption in the inductive dynamics enables correlation in the deliberators' beliefs to emerge spontaneously, resulting in the deliberators converging to correlated equilibrium. ;I devote the majority of the dissertation to the formal theory of inductive deliberation and correlated equilibrium. In particular, I show the following: Under suitable conditions, correlated equilibria correspond to fixed points of the dynamics, and The dynamics can create correlation in beliefs from an initial uncorrelated state. In the final chapter of the dissertation, I give an account of the origins of social conventions such as the use of particular words in human languages as the result of inductive deliberation. (shrink)

The usual justification for Nash equilibrium behavior involves (at least implicitly) the assumption that it is common knowledge among the players both that the Nash equilibrium in question will be played by all and that all players are expected utility maximizers. We show that in a large class of extensive form games, the assumption that rationality is common knowledge cannot be maintained throughout the game. It is shown that these can have serious consequences on traditional extensive form solution concepts (such (...) as Selten's (1965) notion of subgame perfect Nash equilibria). (shrink)

This book explores how individual actions coordinate to produce unintended social consequences. In the past this phenomenon has been explained as the outcome of rational, self-interested individual behaviour. Professor Bicchieri shows that this is in no way a satisfying explanation. She discusses how much knowledge is needed by agents in order to coordinate successfully. If the answer is unbounded knowledge, then a whole variety of paradoxes arise. If the answer is very little knowledge, then there seems hardly any possibility of (...) attaining coordination. The solution to coordination and cooperation is for agents to learn about each other. The author concludes that rationality must be supplemented by models of learning and by an evolutionary account of how social order can persist. (shrink)