Giacomo Bonanno

• University of California, Davis

Building on the analysis of Bonanno (Artificial Intelligence, 2025) we introduce a simple modal logic containing three modal operators: a unimodal belief operator, a bimodal conditional operator and the unimodal global operator. For each AGM axiom for belief revision, we provide a corresponding modal axiom. The correspondence is as follows: each AG...

We provide a semantic characterization of AGM belief contraction based on frames consisting of a Kripke belief relation and a Stalnaker-Lewis selection function. The central idea is as follows. Let K be the initial belief set and K-A be the contraction of K by the formula A; then B belongs to the set K-A if and only if, at the actual state, the age...

We argue in favor of a departure from the equilibrium approach in game theory towards the less ambitious goal of describing only the actual behavior of rational players. The notions of Nash equilibrium and its refinements require a specification of the players’ choices and beliefs not only along the equilibrium play but also at counterfactual histo...

We consider decision problems under complete ignorance and extend the minimax regret principle to situations where, after taking an action, the decision maker does not necessarily learn the state of the world. For example, if the decision maker only learns what the outcome is, then all she knows is that the actual state is one of the (possibly seve...

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 exten...

There is an ongoing debate in the philosophical literature whether the conditionals that are central to deliberation are subjunctive or indicative conditionals and, if the latter, what semantics of the indicative conditional is compatible with the role that conditionals play in deliberation. We propose a possible-world semantics where conditionals...

This textbook incorporates the author’s previous book "The Economics of Uncertainty and Insurance" and extends it with the addition of several new chapters on risk sharing, asymmetric information, adverse selection, signaling and moral hazard. It provides a comprehensive introduction to the analysis of economic decisions under uncertainty and to th...

We consider a basic logic with two primitive uni-modal operators: one for certainty and the other for plausibility. The former is assumed to be a normal operator (corresponding—semantically—to a binary Kripke relation), while the latter is merely a classical operator (corresponding—semantically—to a neighborhood function). We then define belief, in...

In an earlier paper [Rational choice and AGM belief revision, Artificial Intelligence, 2009] a correspondence was established between the choice 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 r...

In an earlier paper [Rational choice and AGM belief revision, Artificial Intelligence, 2009] a correspondence was established between the choice 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 r...

This text provides an introduction to the analysis of economic decisions under uncertainty, with particular focus on insurance markets. "The Economics of Uncertainty and Insurance" is relatively short (220 pages) and richly illustrated with 80 figures. It is suitable for both self-study and as the basis for an upper-division undergraduate course. T...

Doxastic characterizations of the set of Nash equilibrium outcomes and of the set of backward-induction outcomes are provided for general perfect-information games (where there may be multiple backward-induction solutions). We use models that are behavioral, rather than strategy-based, where a state only specifies the actual play of the game and no...

We study common belief of weak-dominance rationality in strategic-form games with ordinal utilities, employing a qualitative model of beliefs. We characterize two standard solution concepts for such games: the Iterated Deletion of Börgers-dominated Strategies (IDBS) and the Iterated Deletion of Inferior Strategy Profiles (IDIP). We do so by imposin...

This textbook provides an introduction to the topic of rational decision making as well as a brief overview of the most common biases in judgment and decision making. "Decision Making" is relatively short (300 pages) and richly illustrated with approximately 100 figures. It is suitable for both self-study and as the basis for an upper-division unde...

In (Bonanno, 2013), a solution concept for extensive-form games, called perfect Bayesian equilibrium (PBE), was introduced and shown to be a strict refinement of subgame-perfect equilibrium; it was also shown that, in turn, sequential equilibrium (SE) is a strict refinement of PBE. In (Bonanno, 2016), the notion of PBE was used to provide a charact...

In (Bonanno, Int J Game Theory 42:567–592, 2013) a general notion of perfect Bayesian equilibrium (PBE) for extensive-form games was introduced and shown to be intermediate between subgame-perfect equilibrium and sequential equilibrium. Besides sequential rationality, the ingredients of the proposed notion are (1) the existence of a plausibility or...

This is an Open Access textbook on non-cooperative Game Theory with 165 solved exercises

In 2011 Harold Camping, president of Family Radio (a California-based Christian radio station), predicted that Rapture (the taking up into heaven of God’s elect people) would take place on May 21, 2011. In light of this prediction some of his followers gave up their jobs, sold their homes and spent large sums promoting Camping’s claims. Did these p...

We investigate an extension of the notion of backward induction to dynamic games with imperfect information and provide a doxastic characterization of it. Extensions of the idea of backward induction were proposed by Penta (2009) and later by Perea (2014), who also provided a doxastic characterization in terms of the notion of common belief of futu...

Johan van Benthem has highlighted in his work that many questions arising in the analysis of strategic interaction call for logical and computational analysis. These questions lead to both formal and conceptually illuminating answers, in that they contribute to clarifying some of the underlying assumptions behind certain aspects of game-theoretical...

This special issue of the Journal of Philosophical Logic contains a selection of papers presented at the 10th Conference on Logic and the Foundations of Game and Decision Theory (LOFT10), which took place at the University of Sevilla (Spain), July 18-20, 2012.

The analysis of rational play in dynamic games is usually done within a static framework that specifies a player's initial beliefs as well as his disposition to revise those beliefs conditional on hypothetical states of information. We suggest a simpler approach, where the rationality of a player's choice is judged on the basis of the actual belief...

This is the first draft of a chapter for the forthcoming Handbook of Epistemic Logic, edited by Hans van Ditmarsch, Joe Halpern, Wiebe van der Hoek and Barteld Kooi (College Publications). Contents: 1. Introduction 2. Epistemic Models of Strategic-Form Games 3. Semantic Analysis of Common Belief of Rationality 4. Syntactic Characterization of Commo...

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'...

We discuss the issues that arise in modeling the notion of common belief of rationality in epistemic models of dynamic games, in particular at the level of interpretation of strategies. A strategy in a dynamic game is defined as a function that associates with every information set a choice at that information set. Implicit in this definition is a...

We provide a general notion of perfect Bayesian equilibrium which can be applied to arbitrary extensive-form games and is intermediate between subgame-perfect equilibrium and sequential equilibrium. The essential ingredient of the proposed definition is the qualitative notion of AGM-consistency, which has an epistemic justification based on the the...

This book constitutes the refereed proceedings of the 8th International Conference on Logic and the Foundations of the Theory of Game and Decision Theory, LOFT8 2008, held in Amsterdam, The Netherlands, July 2008. This volume is based on a selection of the presented papers and invited talks. They survived a thorough and lengthy reviewing process. T...

We study belief change in the branching-time structures introduced in [4]. First, we identify a property of branching-time frames that is equivalent 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 f...

The paradigm for modelling decision-making under uncertainty has undoubtedly been the theory of Expected Utility, which was first developed by von Neumann and Morgenstern (1944) and later extended by Savage (1954) to the case of subjective uncertainty. The inadequacy of the theory of Subjective Expected Utility (SEU) as a descriptive theory was soo...

In [G. Bonanno, Rational choice and AGM belief revision, Artificial Intelligence, 2009] a semantics for one-stage AGM belief revision was pro-posed based on choice frames, borrowed from the rational choice litera-ture. In this paper we extend the semantics of choice frames to deal with iterated belief revision and use the corresponding structures t...

In previous work (G. Bonanno, Rational choice and AGM belief re-vision, Artificial Intelligence, 2009) a semantics for AGM belief revision was proposed based on choice frames, borrowed from the rational choice literature. In this paper we discuss how to use choice frames to ana-lyze extensive-form games. Given an extensive form with perfect recall,...

We establish a correspondence between the rationalizability of choice studied in the revealed preference literature and the notion of minimal belief revision captured by the AGM postulates. A choice frame consists of a set of alternatives Ω, a collection E of subsets of Ω (representing possible choice sets) and a function f:E→Ω2 (representing choic...

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 imag...

This is the first draft of the entry “Game Theory” to appear in the Sage Handbook of the Philosophy of Social Science (edited by Ian Jarvie & Jesús Zamora Bonilla), Part III, Chapter 16.

This volume is a collects papers originally presented at the 7th Conference on Logic and the Foundations of Game and Decision Theory (LOFT), held at the University of Liverpool in July 2006. LOFT is a key venue for presenting research at the intersection of logic, economics, and computer science, and this collection gives a lively and wide-ranging...

It is natural to think of belief revision as the interaction of belief and information over time. Thus branching-time temporal logic seems a natural setting for a theory of belief revision. We propose a logic based on three modal operators: a belief operator, an information operator and a next-time operator. Four logics of increasing strength are p...

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 provi...

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 thi...

his working paper contains the slides of two invited lectures on the Epistemic Foundations of Game Theory, delivered at the Royal Netherlands Academy of Arts and Sciences (KNAW) on February 8, 2007.

Within the class of structures introduced in [G. Bonanno, Axiomatic characterization of the AGM theory of belief revision in a temporal logic, Artificial Intelligence, 171 (2007), 144-160] we consider the subclass satisfying the property that, for every state-instant pair (omega,t), there is an ordering of the set of states that rationalizes the re...

From 26.08. to 30.08.2007, the Dagstuhl Seminar 07351 ``Formal Models of Belief Change in Rational Agents'' was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentation...

Semantic structures for belief revision and iterated belief revision are proposed. We start with one-stage revision structures that generalize the notion of choice function from rational choice theory. A correspondence between these one-stage structures and AGM belief revision functions is established. We then add branching time and consider more g...

From August 26, 2007 to August 30, 2007, the Dagstuhl Seminar 07351 "Formal Models of Belief Change in Rational Agents" was held at the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. The Executive Sum...

We propose a modal logic based on three operators, representing intial beliefs, information and revised beliefs. Three simple axioms are used to provide a sound and complete axiomatization of the qualitative part of Bayes’ rule. Some theorems of this logic are derived concerning the interaction between current beliefs and future beliefs. Informatio...

It is natural to think of belief revision as the interaction of belief and information over time. Thus branching-time temporal logic seems a natural setting for a theory of belief revision. 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 lo...

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 gam...

We propose a logic based on three modal operators, representing intial beliefs, information and revised beliefs. Three simple and transparent axioms are used to provide a sound and complete axiomatization of the qualitative part of Bayes'' rule. Some theorems of this logic are derived concerning the interaction between current beliefs and future be...

The temporal updating of an agent’s beliefs in response to a flow of information is modeled in a simple modal logic that, for every date t, contains a normal belief operator B t and a non-normal information operator I t which is analogous to the ‘only knowing’ operator discussed in the computer science literature. Soundness and completeness of the...

This document gathers the panelists' contribution. @InProceedings{levi_et_al:DSP:2005:358, author = {Isaac Levi and Giacomo Bonanno and Bernard Walliser and Didier Dubois and Hans Rott and James Delgrande and J{'e}r{^o}me Lang}, title = {05321 -- Panel on belief change}, booktitle = {Belief Change in Rational Agents: Perspectives from Artificial In...

The notion of perfect recall in extensive games was introduced by Kuhn [in: Contributions to the Theory of Games, Vol. 2, 1953, p. 193], who interpreted it as “equivalent to the assertion that each player is allowed by the rules of the game to remember everything he knew at previous moves and all of his choices at those moves.” We provide a syntact...

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 gam...

Two notions of memory are studied both syntactically and semantically: memory of past beliefs and memory of past actions. The analysis is carried out in a basic temporal logic framework enriched with beliefs and actions.

We provide a syntactic characterization of the property of perfect recall in extensive games. The language we use is basic temporal logic with the addition of a knowledge operator for every player.

Two views of game theory are discussed: (1) game theory as a description of the behavior of rational individuals who recognize each other's rationality and reasoning abilities, and (2) game theory as an internally consistent recommendation to individuals on how to act in interactive situations. It is shown that the same mathematical tool, namely mo...

The paper models information as possibilities consistent with signals received from the environment. Knowledge is obtained by reasoning about the signals received as well as those that might have been received but were not. The term "knowledge" is used to refer to those beliefs that are obtained by reasoning about the available information, and not...

The logical foundations of game-theoretic solution concepts have so far been explored within the confines of epistemic logic. In this paper we turn to a different 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 age...

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 t...

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 ar...

Curb sets [Basu and Weibull, Econ. Letters 36 (1991), 141-146] are product sets of pure strategies containing all individual best-responses against beliefs restricted to the recommendations to the remaining players. The concept of minimal curb sets is a set-theoretic coarsening of the notion of strict Nash equilibrium. We introduce the concept of m...

The logic of common belief does not always re‡ect that of individual beliefs. In particular, even when the individual belief operators satisfy the KD45 logic, the common belief operator may fail to satisfy axiom 5. That is, it can happen that neither is A commonly believed nor is it common belief that A is not commonly believed. We identify the int...

Recent contributions have questioned the meaningfulness of the Common Prior Assumption (CPA) in situations of incomplete information. We characterize the CPA in terms of the primitives (individuals' belief hierarchies) without reference to an ex ante stage. The key is to rule out "agreeing to disagree" about any aspect of beliefs. Our results also...

We provide a self-contained, selective overview of the literature on the role of knowledge and beliefs in game theory. We focus on recent results on the epistemic foundations of solution concepts, including correlated equilibrium, rationalizability in dynamic games, forward and backward induction.

Curb sets [Basu and Weibull, Econ. Letters 36 (1991), 141-146] are product sets of pure strategies containing all individual best-responses against beliefs restricted to the recommendations to the remaining players. The concept of minimal curb sets is a set-theoretic coarsening of the notion of strict Nash equilibrium. We introduce the concept of m...

The logic of common belief does not always reflect the logic of individual beliefs. In particular, the Negative Introspection property fails at the common belief level, that is, it can happen that neither is A commonly believed nor is it common belief that A is not commonly believed. Indeed Negative Introspection of common belief involves restricti...

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...

The logical foundations of game-theoretic solution concepts have so far been developed within the confines of epistemic logic. In this paper we turn to a different branch of modal logic, namely temporal logic, and propose to view the solution of a game as a complete prediction about future play. We extend the branching time framework by adding agen...

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 (Bertrand) or less intense (Cournot) competition. We find that there are cost-reducing innovations that are pursued under Cournot but not under Bertrand competition. The seco...

Within an incomplete information framework (where the primitives are the individuals’ belief hierarchies) the Truth Axiom is stated locally as the hypothesis that no individual has any false beliefs and that this fact is common belief. We decompose the ‘Truth Condition’ into three parts: Truth of common belief, Truth about common belief and ‘quasi-...

We consider interactive epistemic models where individuals are described by both their "knowledge" and their "beliefs." Three intersubjective consistency conditions are examined: Intersubjective Caution (if an individual believes something to be common belief then he knows it to be common belief), Truth of Common Belief (only true facts are commonl...

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...

We provide two local characterizations of Negative Introspection of Common Belief (NI). The first shows NI to be equivalent to the conjunction of Truth of individuals' belief about what is commonly believed (TCB) and common belief in TCB. According to the second, NI corresponds to a general reducibility property of higher-order beliefs about common...

Aumann (1976) put forward a formal definition of common knowledge and used it to prove that two "like minded" individuals cannot "agree to disagree" in the following sense. If they start from a common prior and update the probability of an event E (using Bayes' rule) on the basis of private information, then it cannot be common knowledge between th...

We provide an introduction to interactive belief systems from a qualitative and semantic point of view. Properties of belief hierarchies are formulated locally. Among the properties considered are "Common belief in no error" (which has been shown to have important game theoretic applications), "Negative introspection of common belief" and "Truth ab...

The interaction bietween knowledge and belief in a temporal context is analyzed. An axiomatic formulation and semantic characterization of the principle of belief persistency implied by the standard conditionalization rule are provided. This principle says that an individual does not change her mind unless new evidence forces her to do so. It is sh...