Giacomo Bonanno

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

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.

Within the context of extensive-form (or dynamic) games, we use choice frames to represent the initial beliefs of a player as well as her disposition to change those beliefs when she learns that an information set of hers has been reached. As shown in [5], in order for the revision operation to be consistent with the AGM postulates [1], the player’s choice frame must be rationalizable in terms of a total pre-order on the set of histories. We consider four properties of choice frames and show that, together with the hypothesis of a common prior, are necessary and sufficient for the existence of a plausibility order that rationalizes the epistemic state (that is, initial beliefs and disposition to revise those beliefs) of all the players. The plausibility order satisfies the properties introduced in [6] as part of a new definition of perfect Bayesian equilibrium for dynamic games. Thus the present paper provides epistemic foundations for that solution concept

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

Counterexamples to two results by Stalnaker (Theory and Decision, 1994) are given and a corrected version of one of the two results is proved. Stalnaker's proposed results are: (1) if at the true state of an epistemic model of a perfect information game there is common belief in the rationality of every player and common belief that no player has false beliefs (he calls this joint condition ‘strong rationalizability’), then the true (or actual) strategy profile is path equivalent to a Nash equilibrium; (2) in a normal-form game a strategy profile is strongly rationalizable if and only if it belongs to C8 , the set of profiles that survive the iterative deletion of inferior profiles

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.

The logical foundations of game-theoretic solution concepts have so far been explored within the con¯nes of epistemic logic. In this paper we turn to a di®erent branch of modal logic, namely temporal logic, and propose to view the solution of a game as a complete prediction about future play. The branching time framework is extended by adding agents and by de¯ning the notion of prediction. A syntactic characterization of backward induction in terms of the property of internal consistency of prediction is given

Since Lewis’s (1969) and Aumann’s (1976) pioneering contributions, the concepts of common knowledge and common belief have been discussed extensively in the literature, both syntactically and semantically1. At the individual level the difference between knowledge and belief is usually identified with the presence or absence of the Truth Axiom ( iA → A), which is interpreted as ”if individual i believes that A, then A is true”. In such a case the individual is often said to know that A (thus it is possible for an individual to believe a false proposition but she cannot know a false proposition). Going to the interpersonal level, the literature then distinguishes between common knowledge and common belief on the basis of whether or not the Truth Axiom is postulated at the individual level. However, while at the individual level the Truth Axiom captures merely a relationship between the individuals’ beliefs and the external world, at the interpersonal level it has very strong implications. For example, the following is a consequence of the Truth Axiom: i jA → iA, that is, if individual i believes that individual j believes that A, then individual i herself believes that A. Thus, in contrast to other axioms, the Truth Axiom does not merely reflect individual agents’ “logic of belief”. (The reason why the Truth Axiom is much stronger in an interpersonal context than appears at first glance is that it amounts to assuming that agreement of any individual’s belief with the truth is common knowledge). Given its logical force, it is not surprising to find that it has strong implications for the logic of common knowledge. In particular, if each individual’s beliefs satisfy the strongest logic of knowledge (namely S5 or KT5), the associated common knowledge operator satisfies this logic too. Such is not the case for belief: bereft of the Truth Axiom, even the strongest logic for individual belief (KD45) is insufficient to ensure the satisfaction of the “Negative Introspection” axiom for common belief: ¬ ∗A → ∗¬ ∗A (where ∗ denotes the common belief operator)..

An information completion of an extensive game is obtained by extending the information partition of every player from the set of her decision nodes to the set of all nodes. The extended partition satisfies Memory of Past Knowledge (MPK) if at any node a player remembers what she knew at earlier nodes. It is shown that MPK can be satisfied in a game if and only if the game is von Neumann (vN) and satisfies memory at decision nodes (the restriction of MPK to a player's own decision nodes). A game is vN if any two decision nodes that belong to the same information set of a player have the same number of predecessors. By providing an axiom for MPK we also obtain a syntactic characterization of the said class of vN games.

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.”

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..

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