Are the Players in an Interactive Belief Model Meta-certain of the Model Itself?

TL;DR

This paper formalizes the concept of common meta-certainty in interactive belief models, linking it to introspective beliefs and showing its role in epistemic characterizations of game solutions.

Contribution

It introduces a formal framework for common meta-certainty, connecting it to introspective beliefs and rationality in game-theoretic models.

Findings

Players are introspective if they are certain of their belief-generating maps.

Common meta-certainty of the model is equivalent to common belief of players' beliefs.

Rationality beliefs lead to actions consistent with iterated elimination of dominated strategies.

Abstract

In an interactive belief model, are the players "commonly meta-certain" of the model itself? This paper formalizes such implicit "common meta-certainty" assumption. To that end, the paper expands the objects of players' beliefs from events to functions defined on the underlying states. Then, the paper defines a player's belief-generating map: it associates, with each state, whether a player believes each event at that state. The paper formalizes what it means by: "a player is (meta-)certain of her own belief-generating map" or "the players are (meta-)certain of the profile of belief-generating maps (i.e., the model)." The paper shows: a player is (meta-)certain of her own belief-generating map if and only if her beliefs are introspective. The players are commonly (meta-)certain of the model if and only if, for any event which some player i believes at some state, it is common belief at the state that player i believes the event. This paper then asks whether the "common meta-certainty" assumption is needed for an epistemic characterization of game-theoretic solution concepts. The paper shows: if each player is logical and (meta-)certain of her own strategy and belief-generating map, then each player correctly believes her own rationality. Consequently, common belief in rationality alone leads to actions that survive iterated elimination of strictly dominated actions.