[Paper Review] From rules to runs: A dynamic epistemic take on imperfect information games

Kai Li, Yanjing Wang|arXiv (Cornell University)|Dec 7, 2015

Game Theory and Applications21 references3 citations

TL;DR

This paper proposes a dynamic epistemic framework that separates game rules from game runs in imperfect information games, enabling step-by-step computation of player knowledge and actions. It introduces a modal logic with complete axiomatization, showing that under structural similarity, mixing rules and runs is not harmful despite conceptual confusion in traditional game theory.

ABSTRACT

In the literature of game theory, the information sets of extensive form games have different interpretations, which may lead to confusions and paradoxical cases. We argue that the problem lies in the mix-up of two interpretations of the extensive form game structures: game rules or game runs which do not always coincide. In this paper, we try to separate and connect these two views by proposing a dynamic epistemic framework in which we can compute the runs step by step from the game rules plus the given assumptions of the players. We propose a modal logic to describe players' knowledge and its change during the plays, and provide a complete axiomatization. We also show that, under certain conditions, the mix-up of the rules and the runs is not harmful due to the structural similarity of the two.

Motivation & Objective

• To resolve conceptual confusion in extensive form games caused by conflating game rules and game runs.
• To formalize the distinction between physical game rules and temporal game runs in imperfect information settings.
• To develop a dynamic epistemic logic that models players' knowledge evolution during game plays.
• To provide a complete axiomatization of the proposed logic for reasoning about knowledge and actions.
• To show that under structural similarity, mixing rules and runs does not lead to errors in reasoning.

Proposed method

• Introduce a dynamic epistemic logic (LDEL) to describe players' knowledge and its change during game plays.
• Define an update product mechanism that computes game runs from rules and player assumptions step by step.
• Model player knowledge using epistemic states derived from the update process, incorporating observational power and memory.
• Use a formal semantics where game runs are generated from initial rules and assumptions via iterative updates.
• Ensure completeness of the logic through a sound and complete axiomatization system.
• Adapt the framework to allow extensions such as non-perfect recall, explicit memory, and higher-order uncertainty.

Experimental results

Research questions

• RQ1 How can game rules and game runs be formally separated in extensive form games with imperfect information?
• RQ2 What logical framework enables dynamic modeling of players' knowledge during game progression?
• RQ3 Under what conditions is the confusion between rules and runs not problematic?
• RQ4 Can a complete axiomatization be provided for a logic modeling knowledge and action in such games?
• RQ5 How can the framework be extended to handle memory, concurrency, and higher-order uncertainty?

Key findings

• The framework successfully separates game rules from game runs, resolving long-standing conceptual ambiguities in extensive form game theory.
• The proposed dynamic epistemic logic (LDEL) is complete, providing a formal system for reasoning about knowledge evolution in imperfect information games.
• The update mechanism computes game runs step-by-step from rules and player assumptions, enabling dynamic knowledge tracking.
• Structural similarity between rules and runs explains why mixing them is not always harmful, even if conceptually distinct.
• The model supports extensions to non-perfect recall, explicit memory, and higher-order uncertainty through modular design.
• The framework enables model checking without full run enumeration, offering computational efficiency for logical verification.