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Area of Science:

  • Theoretical Computer Science
  • Game Theory
  • Stochastic Processes

Background:

  • Stochastic 2-player games with reachability objectives are fundamental in decision-making under uncertainty.
  • Understanding strategy memory requirements is crucial for computational complexity and algorithm design.

Purpose of the Study:

  • To fully characterize the memory needs of epsilon-optimal and optimal strategies in infinite stochastic games.
  • To investigate the impact of action set sizes and uniformity on memory requirements.

Main Methods:

  • Analysis of memory bounds for epsilon-optimal and optimal strategies.
  • Consideration of uniform strategies (independent of the start state).
  • Examination of specific cases like infinitely branching turn-based games.

Main Results:

  • Epsilon-optimal Maximizer strategies necessitate infinite memory if the Minimizer has infinite action sets.
  • Even with guaranteed winning strategies, finite memory (step counter plus private memory) is insufficient in some infinite games.
  • Memoryless uniform epsilon-optimal Maximizer strategies may not exist, even with finite action sets or finitely branching games.

Conclusions:

  • The memory complexity of strategies in infinite stochastic games is highly sensitive to player action sets and uniformity requirements.
  • A single bit of public memory suffices for uniform epsilon-optimal Maximizer strategies in games with finite action sets.