A Logic for Suspicious Players: Epistemic Actions and Belief–Updates in Games

Abstract

This paper introduces a notion of ‘epistemic action’ to describe changes in the information states of the players in a game. For this, ideas are developed from earlier contributions. The ideas are enriched to cover not just purely epistemic actions, but also fact–changing actions (‘real moves’, e.g. choosing a card, exchanging cards etc.) and nondeterministic actions and strategies (conditional actions having knowledge tests as conditions). The author considers natural operations with epistemic actions and uses them to describe significant aspects of the interaction between beliefs and actions in a game. A logic is used that combines in a specific way a multiagent epistemic logic with a dynamic logic of ‘epistemic actions’. The author presents a complete and decidable proof system for this logic. As an application, the author analyses a specific example of a dialogue game (a version of the Muddy Children Puzzle, in which some of the children can ‘cheat’ by engaging in secret communication moves, while others may be punished for their credulity). Also presented is a sketch of a ‘rule–based’ approach to games with imperfect information (allowing ‘sneaky’ possibilities, such as cheating, being deceived and suspecting the others to be cheating).