Jonathan A. Zvesper

Keep ‘hoping’ for rationality: a solution to the backward induction paradox

We formalise a notion of dynamic rationality in terms of a logic of conditional beliefs on (doxastic) plausibility models. Similarly to other epistemic statements (e.g. negations of Moore sentences and of Muddy Children announcements), dynamic rationality changes its meaning after every act of learning, and it may become true after players learn it is false. Applying this to extensive games, we “simulate” the play of a game as a succession of dynamic updates of the original plausibility model: the epistemic situation when a given node is reached can be thought of as the result of a joint act of learning (via public announcements) that the node is reached. We then use the notion of “stable belief”, i.e. belief that is preserved during the play of the game, in order to give an epistemic condition for backward induction: rationality and common knowledge of stable belief in rationality. This condition is weaker than Aumann’s and compatible with the implicit assumptions (the “epistemic openness of the future”) underlying Stalnaker’s criticism of Aumann’s proof. The “dynamic” nature of our concept of rationality explains why our condition avoids the apparent circularity of the “backward induction paradox”: it is consistent to (continue to) believe in a player’s rationality after updating with his irrationality.

Common knowledge in interaction structures

We consider two simple variants of a framework for reasoning about knowledge amongst communicating groups of players. Our goal is to clarify the resulting epistemic issues. In particular, we investigate what is the impact of common knowledge of the underlying hypergraph connecting the players, and under what conditions common knowledge distributes over disjunction. We also obtain two versions of the classic result that common knowledge cannot be achieved in the absence of a simultaneous event (here a message sent to the whole group).

From Lawvere to Brandenburger-Keisler: Interactive Forms of Diagonalization and Self-reference

Diagonal arguments lie at the root of many fundamental phenomena in the foundations of logic and mathematics. Recently, a striking form of diagonal argument has appeared in the foundations of epistemic game theory, in a paper by Adam Brandenburger and H. Jerome Keisler [11]. The core Brandenburger-Keisler result can be seen, as they observe, as a two-person or interactive version of Russell’s Paradox.

A note on assumption-completeness in modal logic

We study the notion of assumption-completeness, which is a property of belief models first introduced in [18]. In that paper it is considered a limitative result - of significance for game theory - if a given language does not have an assumption-complete belief model. We show that there are assumption-complete models for the basic modal language (Theorem 8).

From Lawvere to Brandenburger–Keisler: Interactive forms of diagonalization and self-reference

Abstract We analyze the Brandenburger–Keisler paradox in epistemic game theory, which is a ‘two-person version of Russells paradox’. Our aim is to understand how it relates to standard one-person arguments, and why the ‘believes–assumes’ modality used in the argument arises. We recast it as a fixpoint result, which can be carried out in any regular category, and show how it can be reduced to a relational form of the one-person diagonal argument due to Lawvere. We give a compositional account, which leads to simple multi-agent generalizations. We also outline a general coalgebraic approach to the construction of assumption-complete models.

Distributed iterated elimination of strictly dominated strategies

We characterize epistemic consequences of truthful communication among rational agents in a game-theoretic setting. To this end we introduce normal-form games equipped with an interaction structure, which specifies which groups of players can communicate their preferences with each other. We then focus on a specific form of interaction, namely a distributed form of iterated elimination of strictly dominated strategies (IESDS), driven by communication among the agents. We study the outcome of IESDS after some (possibly all) messages about players’ preferences have been sent. The main result of the paper, Theorem 4, provides an epistemic justification of this form of IESDS.

Dynamics we can believe in: a view from the Amsterdam School on the centenary of Evert Willem Beth

Logic is breaking out of the confines of the single-agent static paradigm that has been implicit in all formal systems until recent times. We sketch some recent developments that take logic as an account of information-driven interaction. These two features, the dynamic and the social, throw fresh light on many issues within logic and its connections with other areas, such as epistemology and game theory.