A logic for suspicious players : epistemic actions and belief-update in games

In this paper, we introduce a notion of ``epistemic action'' to describe changes in the information states of the players in a game. For this, we use ideas that we have developed in our previous papers [BMS], [BMS2] and [B], enriching them 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). We consider natural operations with epistemic actions and we use them to describe significant aspects of the interaction between beliefs and actions in a game. For this, we use a logic that combines in a specific way a multi-agent epistemic logic with a dynamic logic of ``epistemic actions''. We give (without proof) a complete and decidable proof system for this logic. As an application, we analyze 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). We also present 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).