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Pierpaolo Battigalli: Type Structures and Interactive Epistemology in Games

Type structures are widely used in game theory to model interactive beliefs, i.e. the beliefs of the players about the external states as well as their beliefs about each other beliefs. They were first proposed as a tool to represent the epistemic states of players when the rules of the game and/or players’ preferences over consequences are not commonly known (incomplete information), and therefore players form expectations about unknown parameters, the private knowledge of other players and the expectations of other players.

But it was then understood that the same mathematical tools could be used to provide epistemic characterizations of solution concepts. More generally, type structures provided a rigorous language allowing to express assumptions about players’ rationality and interactive beliefs, and then derive the behavioral consequences of such assumptions.

A type of a player in a type structure is mapped to a joint belief about the external states and the types of other players. Thus, type structures provide an implict, self-referential description of interactive beliefs. But it can be shown that that every type corresponds to an infinite hierarchy of beliefs: a first-order belief about the external state, a second-order joint belief about the external state and the first-order beliefs of the other players, etc. Explicit descriptions of interactive beliefs can be obtained by constructing the space of all hierarchies of beliefs. Under regularity conditions it is shown that the subspace of “collectively coherent” hierarchies corresponds to a “canonical” type structure. This structure is complete (for every conceivable belief about the external state and other players, there is a type that holds such belief), and terminal (every other structure can be mapped to the canonical structure in a way that preserves beliefs hierarchies). It is argued that the canonical structure provides an appropriate framework to represent expressible assumptions about rationality and interactive beliefs.

The tutorial will focus on type structures for dynamic games, where beliefs are represented as conditional probability systems. These type structures allow a rigorous discussion and characterization of backward and forward induction reasoning.

• "Strong Belief and Forward Induction Reasoning," Journal of Economic Theory, 106 (2002), 356-391 (with M. Siniscalchi).
• "Interactive Beliefs, Epistemic Independence and Strong Rationalizability," Research in Economics (Ricerche Economiche), 53 (1999), 247-273 (with M. Siniscalchi).
• Brandenburger, A., "The Power of Paradox: Some Recent Developments in Interactive Epistemology," International Journal of Game Theory, 35 (2007), 465-492.
• Siniscalchi, M., “Epistemic Game Theory: Belifs and Types”. The New Palgrave Dictionary of Economics, Palgrave Macmillan.

Pierpaolo Battigalli is Professor at the Department of Economics of Bocconi University. He is the director of the PhD program in Economics at Bocconi University.