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. Footnotes Earlier versions of this paper were presented at the 13th Amsterdam Colloquium at the University of Amsterdam, at a workshop on social norms at Wissenschaftskolleg zu Berlin, and at seminars at Tilburg University and the University of Bristol. We are grateful for comments from participants at those meetings, from two anonymous referees, and from Michael Bacharach, Nick Bardsley, Cristina Bicchieri, Luc Bovens, Simon Grant, David McCarthy, Shepley Orr, Brian Skyrms, Peter Vanderschraaf, Peter Wakker and Jörgen Weibull. Robert Sugden's work was supported by the Leverhulme Trust. (shrink)

We present a new class of models of players’ reasoning in non-cooperative games, inspired by David Lewis's account of common knowledge. We argue that the models in this class formalize common knowledge of rationality in a way that is distinctive, in virtue of modelling steps of reasoning; and attractive, in virtue of being able to represent coherently common knowledge of any consistent standard of individual decision-theoretic rationality. We contrast our approach with that of Robert Aumann, arguing that the former avoids (...) and diagnoses certain paradoxes to which the latter may give rise when extended in particular ways. (shrink)

In this paper, I address two connected issues that arise when one considers a rational agent facing a decision problem. One is whether or not the agent may find that the dictates of rationality are such that they cannot all be followed. For example, one may ask whether or not the requirements on the agent's actions imposed by rationality can conflict in an irreconcilable way, making it impossible to satisfy all of them. Put differently, one may ask whether or not (...) any apparent conflict of this type must in fact be capable of rational resolution. I shall say that an agent who is in a position in which the requirements of rationality cannot all be satisfied faces a feasibility dilemma, and I shall characterize certain conceptions of rationality that differ according to whether or not they admit such a possibility. A second issue concerns the number of options that may be deemed rational in a decision problem. Is rationality sufficiently determinate that it always dictates precisely one choice, or may there be more than one rationally permissible option? Is there anything about rationality itself that guarantees that any of the possible options could rationally be chosen? I shall call this issue – whether the concept of rationality itself places any limits on the number of options that may be deemed rational in a given problem – the numbers problem. (shrink)