[Paper Review] Games on Graphs: From Logic and Automata to Algorithms

Nathanaël Fijalkow, Aiswarya, C.+23 more|arXiv (Cornell University)|May 17, 2023

Game Theory and ApplicationsDecision Sciences277 references8 citations

TL;DR

A comprehensive book introducing infinite-duration games on graphs, their qualitative/quantitative objectives, and algorithmic techniques, including linear-time reachability solutions and attractor-based methods, with extensive chapters on classic and extended game models.

ABSTRACT

The objective of this book is to give a comprehensive presentation of the research field concerned with infinite duration games on graphs. Historically, these game models appeared in the study of automata and logic, and they later became important for program verification and synthesis. They have many more applications, in particular some of the models investigated in this book were introduced and studied in neighbouring research communities such as optimisation, reinforcement learning, model theory, and set theory.

Motivation & Objective

• Define arenas, strategies, and qualitative/quantitative objectives for two-player games on graphs.
• Present linear-time algorithms for reachability and fixed-point methods for qualitative objectives.
• Survey classical and modern algorithmic techniques (attractor/strategy improvement/value iteration) across multiple game models.
• Provide foundational results on determinacy and memory (positionality) across finite and infinite arenas.

Proposed method

• Introduce formal definitions of arenas, strategies, and colouring for objectives.
• Develop linear-time attractor-based algorithms for reachability and fixed-point characterizations for winning regions.
• Present and compare algorithm families: value iteration, strategy improvement, and automata-based methods.
• Extend reachability to qualitative and parity objectives using fixed-point theory and memory considerations.
• Explore reductions and translations between edge/vertex labellings and their impact on algorithm design.

Experimental results

Research questions

• RQ1 How can reachability and qualitative objectives be solved efficiently in two-player graph games?
• RQ2 What are the deterministic and memory (positional) properties of winning strategies across various game models?
• RQ3 How do fixed-point methods and attractor constructions underlie algorithmic solutions for qualitative objectives such as parity and Büchi?
• RQ4 How do labeling conventions (edge vs. vertex) affect reductions and complexity of solving games?
• RQ5 What is the landscape of algorithmic approaches for more complex models (stochastic, concurrent, timed, pushdown, multi-objective) compared to classical reachability games?

Key findings

• Reachability objectives in two-player games are solvable with linear time and space, yielding uniform positional strategies for both players.
• An attractor-based fixed-point algorithm computes winning regions, providing a constructive characterization and ranking of vertices.
• Qualitative objectives can be analyzed via fixed-point theory, with deterministic (positional) strategies emerging in many settings.
• Parity and Büchi objectives can be handled through automata-based reductions and memory considerations, linking to standard automata theory.
• Different algorithm families (value iteration, strategy improvement, separating automata) offer quasi-polynomial or exponential-time performance depending on the model.
• Edge- versus vertex-labelling reductions preserve the essential path colour sequences, enabling flexible modelling choices.