Giacomo Bonanno

Restricting attention to the class of extensive games defined by von Neumann and Morgenstern with the added assumption of perfect recall, we specify the information of each player at each node of the game-tree in a way which is coherent with the original information structure of the extensive form. We show that this approach provides a framework for a formal and rigorous treatment of questions of knowledge and common knowledge at every node of the tree. We construct a particular information partition for each player and show that it captures the notion of maximum information in the sense that it is the finest within the class of information partitions that satisfy four natural properties. Using this notion of “maximum information” we are able to provide an alternative characterization of the meet of the information partitions

In her book Rationality and coordination (Cambridge University Press, 1994) Cristina Bicchieri brings together (and adds to) her own contributions to game theory and the philosophy of economics published in various journals in the period 1987-1992. The book, however, is not a collection of separate articles but rather a homogeneous unit organized around some central themes in the foundations of non-cooperative game theory. Bicchieri’s exposition is admirably clear and well organized. Somebody with a good knowledge of game theory would probably benefit mainly from reading the second part of Chapter 3 (from Section 3.6 onward) and Chapter 4. On the other hand, those who have had little exposure to game theory, would certainly benefit from reading the entire book. I shall begin with an overview of the content of the book and then offer some critical comments on what I consider to be the most important part of it

Two questions are examined within a model of vertical differentiation. The first is whether cost-reducing innovations are more likely to be observed in regimes of more intense or less intense competition. Following Delbono and Denicolo (1990) and Bester and Petrakis (1993) we compare two identical industries that differ only in the regime of competition: Bertrand versus Cournot. Since Cournot competition leads to lower output and higher prices, it can be thought of as a regime of less intense competition. We find that the increase in profits associated with any given cost reduction is higher in the case of Cournot competition than in the case of Bertrand competition. Thus there are cost-reducing innovations that would be pursued under Cournot competition but not under Bertrand competition

This paper suggests a way of formalizing the amount of information that can be conveyed to each player along every possible play of an extensive game. The information given to each player i when the play of the game reaches node x is expressed as a subset of the set of terminal nodes. Two definitions are put forward, one expressing the minimum amount of information and the other the maximum amount of information that can be conveyed without violating the constraint represented by the information sets. Our definitions provide intuitive characterizations of such notions as perfect recall, perfect information and simultanetty.

Making a prediction is essentially expressing a belief about the future. It is therefore natural to interpret later predictions as revisions of earlier ones and to investigate the notion of belief revision in this context. We study, both semantically and syntactically, the following principle of minimum revision of prediction: “as long as there are no surprises, that is, as long as what actually occurs had been predicted to occur, then everything which was predicted in the past, if still possible, should continue to be predicted, and no new predictions should be added.”

In an earlier paper [Rational choice and AGM belief revision, _Artificial Intelligence_, 2009] a correspondence was established between the set-theoretic structures of revealed-preference theory (developed in economics) and the syntactic belief revision functions of the AGM theory (developed in philosophy and computer science). In this paper we extend the re-interpretation of those structures in terms of one-shot belief revision by relating them to the trichotomous attitude towards information studied in Garapa (Rev Symb Logic, 1–21, 2020) where information may be either (1) fully accepted or (2) rejected or (3) taken seriously but not fully accepted. We begin by introducing the syntactic notion of _filtered belief revision_ and providing a characterization of it in terms of a mixture of both AGM revision and contraction. We then establish a correspondence between the proposed notion of filtered belief revision and the above-mentioned set-theoretic structures, interpreted as semantic _partial_ belief revision structures. We also provide an interpretation of the trichotomous attitude towards information in terms of the degree of implausibility of the information.

The notion of Nash equilibrium in static oligopoly games is based on the assumption that each firm knows its entire demand curve (and, therefore, its entire profit function). It is much more likely, however, that firms only have some idea of the outcome of small price variations within some relatively small interval of prices. This is because firms can only learn their demand functions through price experiments and if they are risk-averse and/or have a low discount factor, they will be unwilling to engage in extensive price experiments involving large variations in price. We can therefore expect firms to experiment through small price variations and stop when they reach a price such that no small deviation..

When we make a prediction we select, among the conceivable future descriptions of the world, those that appear to us to be most plausible. We capture this by means of two binary relations, ≺c and ≺p: if t1 and t2 are points in time, we interpret t1 ≺ct2 as sayingthat t2 is in the conceivable future of t1, while t1 ≺pt2 is interpreted to mean that t2 isin the predicted future of t1. Within a branching-time framework we propose the following notion of “consistency of prediction”. Suppose that at t1 some future moment t2 is predicted to occur, then every moment t on the unique path from t1 to t2 should also be predicted at t1 and the prediction of t2 should continue to hold at every such t. A sound and complete axiomatization is provided