Giacomo Bonanno

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.

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.

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

We study belief change in the branching-time structures introduced in Bonanno (Artif Intell 171:144–160, 2007 ). First, we identify a property of branching-time frames that is equivalent (when the set of states is finite) to AGM-consistency, which is defined as follows. A frame is AGM-consistent if the partial belief revision function associated with an arbitrary state-instant pair and an arbitrary model based on that frame can be extended to a full belief revision function that satisfies the AGM postulates. Second, we provide a set of modal axioms that characterize the class of AGM-consistent frames within the modal logic introduced in Bonanno (Artif Intell 171:144–160, 2007 ). Third, we introduce a generalization of AGM belief revision functions that allows a clear statement of principles of iterated belief revision and discuss iterated revision both semantically and syntactically.

The principle of belief persistence, or conservativity principle, states that ’\Nhen changing beliefs in response to new evidence, you should continue to believe as many of the old beliefs as possible' (Harman, 1986, p. 46). In particular, this means that if an individual gets new information, she has to accommodate it in her new belief set (the set of propositions she believes), and, if the new information is not inconsistent with the old belief set, then (1) the individual has to maintain all the beliefs she previously had and (2) the change should be minimal in the sense that every proposition in the new belief set must be deducible from the union of the old belief set and the new information (see, e.g., Gardenfors, 1988; Stalnaker, 1984). We focus on this minimal notion of belief persistence and characterize it both semantically and syntactically. A ’possible world' semantic formalization of the principle easily comes to mind. The set of all the propositions that the individual believes corresponds to the set of states of the world that she considers possible and is a subset of the set of states that are not ruled out by the individual's information (or knowledge). It is required that, if the individual considers a state possible and her new information does not exclude this state, then she continue to consider it possible. Furthermore, if the individual regards a particular state as impossible, then she should continue to regard it as impossible unless her new information excludes all the states that she previously regarded as possible. This is closely related to the..

Since belief revision deals with the interaction of belief and information over time, branching-time temporal logic seems a natural setting for a theory of belief change. We propose two extensions of a modal logic that, besides the next-time temporal operator, contains a belief operator and an information operator. The first logic is shown to provide an axiomatic characterization of the first six postulates of the AGM theory of belief revision, while the second, stronger, logic provides an axiomatic characterization of the full set of AGM postulates. © 2006 Elsevier B.V. All rights reserved.

Given an extensive game, with every node x and every player i a subset ki(x) of the set of terminal nodes is associated, and is given the interpretation of player i's knowledge (or information) at node x. A belief of player i is a function that associates with every node x an element of the set Ki(x). A belief system is an n-tuple of beliefs, one for each player. A belief system is rational if it satisfies some natural consistency properties. The main result of the paper is that the notion of rational belief system gives rise to a refinement of the notion of subgame-perfect equilibrium.

ABSTRACT The past fifteen years or so have witnessed considerable progress in our understanding of how the human brain works. One of the objectives of the fast-growing field of neuroscience is to deepen our knowledge of how the brain perceives and interacts with the external world. Advances in this direction have been made possible by progress in brain imaging techniques and by clinical data obtained from patients with localized brain lesions. A relatively new field within neuroscience is neuroeconomics, which focuses on individual decision making and aims to systematically classify and map the brain activity that correlates with decision-making that pertains to economic choices. Neuroeconomic studies rely heavily on functional magnetic resonance imaging (fMRI), which measures the haemodynamic response (that is, changes in the blood flow) related to neural activity in the brain.

The theory of belief revision deals with (rational) changes in beliefs in response to new information. In the literature a distinction has been drawn between belief revision and belief update (see [6]). The former deals with situations where the objective facts describing the world do not change (so that only the beliefs of the agent change over time), while the letter allows for situations where both the facts and the doxastic state of the agent change over time. We focus on belief revision and propose a temporal framework that allows for iterated revision. We model the notion of “minimal” or “conservative” belief revision by considering logics of increasing strength. We move from one logic to the next by adding one or more axioms and show that the corresponding logic captures more stringent notions of minimal belief revision. The strongest logic that we propose provides a full axiomatization of the well-known AGM theory of belief revision.

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.

Adapting a definition introduced by Milgrom (1981) we say that a signal about the environment is good news relative to some initial beliefs if the posterior beliefs dominate the initial beliefs in the sense of first-order stochastic dominance (the assumption being that higher values of the parameter representing the environment mean better environments). We give an example where good news leads to the adoption of a more pessimistic course of action (we say that action a, reveals greater pessimism than action a„ if it gives higher payoff in bad environments and lower payoff in good environments). We then give sufficient conditions for a signal not to induce a more pessimistic choice of action.

We consider strategic-form games with ordinal payoffs and provide a syntactic analysis of common belief/knowledge of rationality, which we define axiomatically. Two axioms are considered. The first says that a player is irrational if she chooses a particular strategy while believing that another strategy is better. We show that common belief of this weak notion of rationality characterizes the iterated deletion of pure strategies that are strictly dominated by pure strategies. The second axiom says that a player is irrational if she chooses a particular strategy while believing that a different strategy is at least as good and she considers it possible that this alternative strategy is actually better than the chosen one. We show that common knowledge of this stronger notion of rationality characterizes the restriction to pure strategies of the iterated deletion procedure introduced by Stalnaker (1994). Frame characterization results are also provided.

The Common Prior Assumption (CPA) plays an important role in game theory and the economics of information. It is the basic assumption behind decision-theoretic justifications of equilibrium reasoning in games (Aumann, 1987, Aumann and Brandenburger, 1995) and no-trade results with asymmetric information (Milgrom and Stokey, 1982). Recently several authors (Dekel and Gul, 1997, Gul, 1996, Lipman, 1995) have questioned whether the CPA is meaningful in situations of incomplete information, where there is no ex ante stage and where the primitives of the model are the individuals' beliefs about the external world (their first-order beliefs), their beliefs about the other individuals' beliefs (second-order beliefs), etc., i.e. their hierarchies of beliefs. In this context, the CPA is a mathematical property whose conceptual content is not clear. The main results of this paper (Theorems 1 and 2) provide a characterization of Harsanyi consistency in terms of properties of the belief hierarchies that are entirely unrelated to the idea of an ex ante stage.