[Paper Review] Games with recurring certainty

Dietmar Berwanger, Anup Basil Mathew|arXiv (Cornell University)|Apr 3, 2014

Formal Methods in Verification11 references4 citations

TL;DR

This paper introduces the concept of 'recurring certainty' in n-player games with imperfect information, where players periodically regain full knowledge of the game state. It proves that such games have decidable winner determination and effective strategy synthesis, leveraging finite tracking via periodic certainty and homomorphic equivalence of epistemic models.

ABSTRACT

Infinite games where several players seek to coordinate under imperfect information are known to be intractable, unless the information flow is severely restricted. Examples of undecidable cases typically feature a situation where players become uncertain about the current state of the game, and this uncertainty lasts forever. Here we consider games where the players attain certainty about the current state over and over again along any play. For finite-state games, we note that this kind of recurring certainty implies a stronger condition of periodic certainty, that is, the events of state certainty ultimately occur at uniform, regular intervals. We show that it is decidable whether a given game presents recurring certainty, and that, if so, the problem of synthesising coordination strategies under w-regular winning conditions is solvable.

Motivation & Objective

• To address the undecidability of distributed synthesis in games with imperfect information, especially when knowledge hierarchies grow unboundedly.
• To identify a natural condition—recurring certainty—that prevents infinite uncertainty and restores decidability.
• To prove that games with recurring certainty admit finite tracking, enabling effective strategy synthesis.
• To show that periodic certainty (bounded intervals of uncertainty) is a consequence of recurring certainty in finite-state games.
• To establish decidability of the winner determination and strategy synthesis problems under recurring certainty

Proposed method

• Introduces the notion of 'recurring certainty' as a weakening of common knowledge, ensuring players regain full state knowledge at regular intervals.
• Proves that in finite-state games, recurring certainty implies periodic certainty—bounded, uniform intervals of state certainty.
• Uses the tracking construction from [5] to model epistemic states, where each node represents a history and its associated knowledge structure.
• Applies homomorphic equivalence to collapse equivalent epistemic models, ensuring finite tracking when uncertainty intervals are bounded.
• Leverages the bounded growth of epistemic models over certainty periods to show finite tracking, even with imperfect information.
• Reduces the original game to a perfect-information two-player game via tracking, enabling decidability and strategy synthesis

Experimental results

Research questions

• RQ1 Can we decide whether a given n-player game with imperfect information exhibits recurring certainty?
• RQ2 Does the existence of recurring certainty imply finite tracking in the epistemic model construction?
• RQ3 Is the problem of determining whether the grand coalition has a winning strategy decidable in games with recurring certainty?
• RQ4 Can finite-state winning strategies be effectively synthesized in such games?
• RQ5 What structural properties (e.g., periodic certainty) emerge from the recurring certainty condition?

Key findings

• Recurring certainty in finite-state games implies periodic certainty, meaning state certainty occurs at uniformly bounded intervals.
• The tracking of games with recurring certainty is finite, due to bounded growth of epistemic models over certainty periods.
• The winner determination problem is decidable for games with recurring certainty and ω-regular winning conditions.
• Finite-state winning strategies can be effectively synthesized for the grand coalition in such games.
• The epistemic models generated during tracking are homomorphically equivalent to singleton models at points of certainty, enabling model compression.
• The undecidability of general imperfect-information games is avoided under the recurring certainty condition due to bounded knowledge hierarchy growth