# Blog Archives

# Topic Archive: Epistemic Game Theory

# Sub-Kripkean epistemic models I

Kripke models provide a versatile semantical tool in modal logic. It appears that when Kripkean methods were adopted in formal epistemology “as is” a key hidden assumption has been overlooked: the very truth condition of a knowledge assertion presupposes that the model itself is common knowledge among the agents. Naturally, this covers only well-controlled epistemic scenarios and leaves off scope numerous situations with partial/asymmetric knowledge. In this talk, we

1) offer a theory of sub-Kripkean epistemic models for unimodal logics;

2) discuss research directions for multi-agent cases.

# Dark Matter of Epistemology

We have already shown that only deductively complete epistemic scenarios admit traditional single-model characterizations. In this talk we show that the paradigmatic Muddy Children story is deductively complete and fairly represented by its standard Kripke model. However, the whole variety of its modifications with partial knowledge, asymmetric knowledge, etc., are not deductively complete and hence are invisible “Dark Matter” from the traditional single-model perspective. We show several examples of such “invisible” scenarios that admit natural syntactic analysis and resolution. Finally, we will discuss a version of Muddy Children, in which justifications become a key ingredient of the solution. These examples represent, in a nutshell, the corresponding classes of real world epistemic problems which lie off limits of the traditional single-model analysis but can be analyzed by a proper combination of syntactic and semantic methods.

# Towards Paraconsistent Game Theory

In this programmatic talk, I will talk about three major examples of paraconsistency and paradoxes in games. Paraconsistent logic is a family of non-classical logics in which contradictions do not entail everything. In this talk, first, I will discuss game theoretical semantics for paraconsistent logic and observe how semantic games change in different, especially in paraconsistent logics. Then, I will consider a self-referential paradox of epistemic game theory, called Brandenburger-Keisler Paradox, and present a model for it. Following, I will shift my attention non-self-referential paradoxes and suggest a non-self-referential paradox in games. These three major cases, I will argue, will be a call for the necessity of the use of non-classical logics in game theoretical reasoning.

# Does knowledge of the model matter?

As we have already discussed, the traditional epistemic reading of Kripke models relies on the hidden assumption of the knowledge of the model by the agent. In the multi-agent setting, e.g. in Epistemic Game Theory, this becomes the assumption of the common knowledge of the model, which is not regarded realistic, and often is theoretically impossible. We sketch logician’s recommendations of how to handle epistemic scenarios without relying on common knowledge of the model.

# Epistemic Considerations on Extensive-Form Games

In this talk, we discuss several topics in extensive-form games. First, we consider perfect information games with belief revision with players who are tolerant of each other’s “hypothetical” errors. Second, we study imperfect information games with players who have no way of assigning probabilities to various possible outcomes, and we analyze how players might choose in such games. We then look into how a truthful manipulator whose motives are not known to players can create a certain knowledge situation about a game, and change the way the game will be played.

# Syntactic View of Strategic Games

We suggest an epistemic logic analysis of strategic games with ordinal payoffs, including Nash’s justification of the equilibrium solution concept: the solution is derived by the players from the game description. We show that most of the classes of games do not have Nash solutions, and for those that have, stronger solution concepts (e.g, domination) and/or refinements of Aumann rationality are needed.

# Bi-simulation of epistemic structures: a semantic proof of completeness for Muddy Children Puzzle.

In this talk we discuss truth preserving bi-simulations of multi-agent Kripke models. As an application, we present a semantic proof of the completeness of the syntactic formulation of Muddy Children Puzzle with respect to its standard model.

# Reasoning about reasoning about games

A mixture of propositional dynamic logic and epistemic logic that we call PDL + E is used to give a formalization of Artemov’s knowledge based reasoning approach to game theory. Epistemic states of players are represented explicitly and reasoned about formally. We give a detailed analysis of the Centipede game using both syntactic proofs and semantical arguments. Formal proofs are used establish what is the case and what is needed to show this. Semantical machinery is used to establish what cannot be the case. All this makes a case that PDL + E can be a useful basis for the logical investigation of game theory.

# Syntactic Epistemic Logic and Games IV

In this talk we consider knowledge structures used to model epistemic components of extensive-form games. We show that the traditional semantic mode of specification by a single model cannot capture some natural epistemic notions, e.g., mutual knowledge of rationality, used in Epistemic Game Theory. In contrast, Syntactic Epistemic Logic handles this and many other conditions that lie outside the scope of the semantic tradition. We outline general principles of Syntactic Epistemic Logic in connection to Game Theory.

# Syntactic Epistemic Logic and Games I

We argue that the traditional way to specify an epistemic situation via a single model is unnecessarily restrictive: there are numerous scenarios which do not have conventional single-model characterizations: less-than-common knowledge (e.g., mutual knowledge), partial, asymmetric knowledge, etc. Syntactic Epistemic Logic, SEL, suggests a way out of this predicament by viewing an epistemic scenario as specified syntactically by a set of formulas which corresponds to the characterization via a class of models rather than a single model. In addition, in Game Theory, SEL helps to seal the conceptual and technical gap, identified by R.Aumann, between the syntactic character of game descriptions and the predominantly semantical way of analyzing games.

This will be the first in the series of talks on SEL. We will consider natural examples of epistemic or game theoretic scenarios which have manageable syntactic description but do not have acceptable single model characterizations.

# Towards Syntactic Epistemic Game Theory

The traditional semantic approach to consider a game as an Aumann structure, though flexible and convenient, is not foundationally satisfactory due to assumptions that a given Aumann structure adequately represents the game and that this structure itself is common knowledge for the players.

These assumptions leave a gap between the officially syntactic character of the game description that often admits multiple models and studying a game as a specific model that is somehow assumed to be commonly known. This gap has been largely ignored or covered up by using as examples simple epistemic scenarios with natural models that were tacitly used as definitions of the game instead of declared syntactic game descriptions. Among others, Aumman found this foundationally unsatisfactory and argued for using what he called ‘Syntactic epistemic logic’ for reasoning about games.

In this talk, we outline a systematic approach to epistemic game theory which we suggest calling ‘Syntactic Epistemic Game Theory’, SEGT, consistent with Aumann’s suggestions, that studies games as they are normally described, in their syntactic form. In SEGT, semantic methods should be properly justified from the original game description. As a case study, we offer a SEGT theory of definitive solutions of strategic games with ordinal payoffs.