Randomized Strategies in Regular Games with Imperfect Information Are Challenging

Core Concepts

Regular games with imperfect information, despite their nice structural properties and the existence of a finite bisimulation in their information tree, do not admit a similar reduction for the synthesis of randomized strategies as in the case of partial-observation games à la Reif. The synthesis problem for randomized strategies in regular games is decidable for reachability and Büchi objectives, but the reductions that worked in games à la Reif no longer hold.

Abstract

The paper considers two-player turn-based games with imperfect information and the synthesis of randomized strategies that ensure the objective is satisfied almost-surely (with probability 1), regardless of the strategy of the other player.
The key insights are:

Regular games with imperfect information, despite their nice structural properties and the existence of a finite bisimulation in their information tree, do not admit a similar reduction for the synthesis of randomized strategies as in the case of partial-observation games à la Reif.

The authors present an algorithmic solution for the synthesis problem with randomized strategies in regular games with reachability objectives. The solution exploits the properties of rectangular morphisms to define a fixpoint computation.

The algorithm is extended to Büchi objectives using reductions from the literature.

The authors show that the reductions that worked in games à la Reif for arbitrary objectives no longer hold in the more general setting of regular games.

The synthesis problem for randomized strategies in regular games is decidable for reachability and Büchi objectives, but the authors leave the decidability status open for safety and coBüchi objectives.

Stats

We consider two-player turn-based games with imperfect information and the synthesis of randomized strategies that ensure the objective is satisfied almost-surely (with probability 1), regardless of the strategy of the other player.
Regular games with imperfect information admit a rectangular morphism, which guarantees that the information tree has a finite bisimulation quotient.

Quotes

"Regular games with imperfect information are not that regular"
"Despite their nice structural properties and the existence of a finite bisimulation in their information tree, regular games are not as well-behaved as games à la Reif."

Key Insights Distilled From

Regular Games with Imperfect Information Are Not That Regular

by Laurent Doye... at arxiv.org 04-01-2024

https://arxiv.org/pdf/2403.20133.pdf

Deeper Inquiries

What are the key structural differences between regular games and games à la Reif that lead to the challenges in synthesizing randomized strategies in regular games

The key structural differences between regular games and games à la Reif that pose challenges in synthesizing randomized strategies in regular games stem from the nature of imperfect information modeling. In regular games, imperfect information is represented by an indistinguishability relation that describes pairs of histories that the player cannot differentiate. This generalization of the traditional model with partial observations in games à la Reif introduces complexities in transferring strategies between games. Unlike in games à la Reif, where a bijection can be established between the original game and the induced abstract game, regular games do not have a direct correspondence that allows for easy transfer of strategies. The unbounded branching degree in the information tree of regular games further complicates the synthesis of randomized strategies, as strategies need to be adapted to account for the potential divergence in information sets.

Can the techniques developed for regular games be extended to handle safety and coBüchi objectives, or are there fundamental limitations

The techniques developed for regular games can be extended to handle safety and coBüchi objectives, but there are fundamental limitations that need to be considered. While the synthesis of randomized strategies for reachability objectives in regular games has been shown to be decidable, the same may not hold true for safety and coBüchi objectives. The undecidability of almost-sure winning strategies for coBüchi objectives in probabilistic automata suggests that the synthesis problem for randomized strategies in regular games with coBüchi objectives may also be undecidable. This fundamental limitation arises from the complexity introduced by the nature of the objectives and the interactions between the player and the environment in games with imperfect information.

How do the complexities of the synthesis problems compare between regular games and games à la Reif, and what are the implications for practical applications

The complexities of the synthesis problems differ between regular games and games à la Reif, with implications for practical applications. In regular games, the synthesis problem for randomized strategies has been shown to be decidable for reachability objectives, with a quadratic algorithmic solution that is feasible for practical implementation. On the other hand, in games à la Reif, the synthesis problem with randomized strategies is undecidable for ω-regular objectives, posing challenges for practical applications requiring complex strategies. The implications of these complexities are significant for the design and analysis of reactive systems, multi-agent systems, and other applications in computer science where decision-making under imperfect information is crucial.