Strategy Complexity of B""uchi Objectives in Concurrent Stochastic Games

2024-04-23

arXiv.org

AI Summary

This paper investigates the memory complexity of strategies for the Max player in two-player zero-sum concurrent stochastic Büchi games over countable graphs, aiming to maximize the probability of visiting a target set infinitely often. We establish that, for any countable graph, ε-optimal strategies require only one bit of public memory plus a step counter—a bound that is tight. Furthermore, we show that this single-bit memory suffices even for the conjunction of Büchi and Transience objectives, whereas finite-memory or memoryless strategies are insufficient. In contrast, for the pure Transience objective on infinite graphs, memoryless ε-optimal strategies exist. Our results settle the exact upper bound on strategy memory complexity for Büchi objectives and yield stronger characterizations for combined objectives. The analysis integrates techniques from game theory, stochastic processes, and strategy synthesis.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Adversarial Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

We study 2-player concurrent stochastic B""uchi games on countable graphs. Two players, Max and Min, seek respectively to maximize and minimize the probability of visiting a set of target states infinitely often. We show that there always exist $varepsilon$-optimal Max strategies that use just a step counter plus 1 bit of public memory. This upper bound holds for all countable graphs, but it is a new result even for the special case of finite graphs. The upper bound is tight in the sense that Max strategies that use just a step counter, or just finite memory, are not sufficient even on finite game graphs. The upper bound is a consequence of a slightly stronger new result: $varepsilon$-optimal Max strategies for the combined B""uchi and Transience objective require just 1 bit of public memory (but cannot be memoryless). Our proof techniques also yield a closely related result, that $varepsilon$-optimal Max strategies for the Transience objective alone (which is only meaningful in infinite graphs) can be memoryless.

Problem

• Study optimal strategies in concurrent stochastic Büchi games
• Maximize probability of visiting target states infinitely often
• Prove ε-optimal strategies need minimal memory (1 bit)

Innovation

• Uses step counter and 1-bit memory
• Optimal for Buechi and Transience objectives
• Memoryless strategies for Transience objective