I am broadly interested in issues at the intersection of logic (especially modal logic), formal epistemology, game theory and social choice theory. My research is currently focused on issues in interactive epistemology, developing mathematical models of group decision making, and on weak systems of modal logic. See my publications page for more details.

#### Interactive Epistemology: Strategic Reasoning in Games

A crucial assumption underlying any game-theoretic analysis is that there is common knowledge that all the players are rational. Rationality here is understood in the decision-theoretic sense: The players' choices are optimal according to some choice rule (for example, maximizing subjective expected utility). Recent work in epistemic game theory has focused on developing sophisticated mathematical models to study the implications of assuming that all the players are rational and this is commonly known (or commonly believed).See my recent entry in the Stanford Encyclopedia of Philosophy and Epistemic Game Theory (the link is given below) for a overview of this area of research. However, if common knowledge of rationality is to have an "explanatory" role in the analysis of a game-theoretic situation, then it is not enough to simply assume that it has obtained in a game situation. It is also important to describe how the players were able to arrive at this crucial state of information.

This research project is focused on analyzing games in terms of the "process of deliberation" that leads the players to select their component of a rational outcome. The goal is to develop a philosophically interesting framework that can represent the players' process of deliberation in game situations and to show how explicitly representing the players' process of deliberation can shed some light on the role that higher-order information (e.g., beliefs about the other players' beliefs) plays in the analysis of game situations. The main challenge is to find the right balance between descriptive accuracy and normative relevance. While this is true for all theories of individual decision making and reasoning, focusing on game situations raises a number of compelling issues. Consult the following articles for highlights of this research project:

#### Interactive Epistemology: Logics of Rational Interaction

Philosophers and computer scientists have developed a variety of logical systems that can be used to reason about communities of (rational and not-so rational) agents engaged in some form of social interaction. Much of this work is focused on developing useful logical frameworks that incorporate insights and ideas from philosophy (especially epistemology and action theory), game theory, and decision theory. The result is a web of logical systems each addressing different aspects of rational agency and social interaction. My research in this area has focused on the central conceptual and technical issues that drive these logical analyses. The overall objective is to see the various logical systems as a coherent account of rational agency and social interaction. There are three main questions:

1. How should we compare different logical frameworks addressing similar aspects of rational agency and social interaction (eg., how information evolves through social interaction)?
2. How should we combine logical systems which address different aspects of social interaction towards the goal of a comprehensive (formal) theory of rational agency?
3. How does a logical analysis contribute to the broader discussion of rational agency and social interaction within philosophy and the social sciences?
Consult the following articles for highlights of this research project:

#### Methods of Group Decision Making

Much of our daily life is spent taking part in various types of what we might call "political" procedures. Examples range from voting in a national election to deliberating with others in small committees to "fairly" dividing a limited resource among a group of people. Many interesting philosophical and mathematical issues arise when we carefully examine our group decision-making processes. Mathematicians, Philosophers and Political Scientists have analyzed in detail many different political procedures, such as fair division algorithms and voting procedures. These analyses typically focus on comparing the mathematical properties of the various group decision-making methods. My ongoing research in this area is focused largely on going beyond studying mathematical properties of the methods used to make a decision by taking into account the many complex phenomena that arise when people taking part in a group decision interact. Consult the following articles for highlights of this research project:

#### Neighborhood Semantics for Modal Logic

Neighborhood models are generalizations of the standard relational models for modal logic invented independently by Dana Scott and Richard Montague in 1970. The underlying idea for neighborhood models comes from point-set topology: In a topology, a neighborhood of a point $$x$$ is any set $$A$$ containing $$x$$ such that you can "wiggle" $$x$$ without leaving $$A$$. What exactly it means to "wiggle" $$x$$ depends on the underlying topology. Using this idea, a general definition for truth of a modal formula $$\Box\varphi$$ runs as follows: $$\Box\varphi$$ is true at a state $$w$$ just in case the truth set of $$\varphi$$ (i.e., the set of states where $$\varphi$$ is true) is a neighborhood of $$w$$. Different semantics for modal logic can be classified in terms of what it means to be a "neighborhood" of a state. For example, in the topological semantics of McKinsey and Tarski, a neighborhood is defined with respect to some given topology. In the standard relational semantics, a neighborhood of $$w$$ is any set containing the states that are immediately connected to $$w$$ via some relation $$R$$ on the set of states $$W$$. The most general approach are neighborhood models in which a neighborhood of $$w$$ is any element of a distinguished collection of sets (the may not necessarily contain the state $$w$$).

A general criticism of neighborhood models is that they are not well-motivated: It is ad hoc to simply assert that certain subsets and not others are in the neighborhood of a given state. They do provide a semantics for weak systems of modal logic, but do they do so in a principled way? There is certainly some truth to this criticism. Nonetheless, recent work has demonstrated the usefulness and interest of neighborhood semantics.

• Neighborhood models naturally show up when studying cooperative and non-cooperative game theory.
• Neighborhood models can be given an epistemic interpretation in which the neighborhoods describe the evidence accepted by a rational (or not so rational) agent.
• Neighborhood models offer an interesting new interpretation of the Barcan and Converse Barcan formulas in first-order modal logic.
Finally, one can learn a great deal about normal systems of modal logic by looking at how these systems behave in a more general semantics. Consult the following articles for highlights of this research project: